Estimation for change point of discretely observed ergodic diffusion processes
EESTIMATION FOR CHANGE POINT OF DISCRETELY OBSERVED ERGODICDIFFUSION PROCESSES
YOZO TONAKI, YUSUKE KAINO, AND MASAYUKI UCHIDA
Abstract.
We treat the change point problem in ergodic diffusion processes from discrete observations.Tonaki et al. (2020) proposed adaptive tests for detecting changes in the diffusion and drift parameters inergodic diffusion models. When any changes are detected by this method, the next question to be consideredis where the change point is. Therefore, we propose the method to estimate the change point of the parameterfor two cases: the case where there is a change in the diffusion parameter, and the case where there is nochange in the diffusion parameter but a change in the drift parameter. Furthermore, we present rates ofconvergence and distributional results of the change point estimators. Some examples and simulation resultsare also given. Introduction
We consider a d -dimensional diffusion process { X t } t ≥ satisfying the stochastic differential equation: (cid:40) d X t = b ( X t , β )d t + a ( X t , α )d W t ,X = x , (1.1)where parameter space Θ = Θ A × Θ B , which is a compact convex subset of R p × R q , θ = ( α, β ) ∈ Θ isan unknown parameter and { W t } t ≥ is a d -dimensional standard Wiener process. The diffusion coefficient a : R d × Θ A −→ R d ⊗ R d and the drift coefficient b : R d × Θ B −→ R d are known except for the parameter θ ,and the true parameter θ ∗ = ( α ∗ , β ∗ ) belongs to Int Θ. We assume that the solution of (1.1) exists, and P θ and E θ denote the law of the solution and the expectation with respect to P θ , respectively. Let { X t i } ni =0 bediscrete observations, where t i = t ni = ih n , and { h n } is a positive sequence with h = h n −→ T = nh −→ ∞ and nh −→ n → ∞ .We deal with the change point problem of parameters in diffusion process models. The change pointproblem consists of two parts: detection of the parameter change and estimation of the change point. Forexample, suppose that we obtain the paths shown in Figure 1. Given such data, the first thing we wouldconsider is whether any changes occur in that path. The paths in Figure 1 clearly shows that some changeoccurs, but the paths in Figure 2 may be difficult to find a change. The first step of the change point problemis to investigate whether there is a change in the data or not, which is “detection of the parameter change”.Suppose that we can detect a change in the paths in Figure 1 or 2. In Figure 1, we can visually see that thechange occurs at t = 60 in the left figure and at t = 40 in the right figure, but the paths in Figure 2 are noteven visually recognizable, so we have no idea where the change occurs. The goal of the change point problemis to investigate the change point when we see that there is a change in the data, which is“estimation of thechange point”.The change point problem for diffusion processes from discrete observations has been studied by manyresearchers. For non-ergodic diffusion processes, see De Gregorio and Iacus (2008) and Iacus and Yoshida(2012). For ergodic diffusion processes, we can refer to Song and Lee (2009), Lee (2011), Negri and Nishiyama(2017), Song (2020), and Tonaki et al. (2020). De Gregorio and Iacus (2008) considered the detection ofthe parameter change in diffusion parameter and the estimation of the change point, and Iacus and Yoshida (Y. Tonaki) Graduate School of Engineering Science, Osaka University, 1-3, Machikaneyama, Toyonaka, Osaka,560-8531, Japan (Y. Kaino)
Graduate School of Engineering Science, Osaka University, 1-3, Machikaneyama, Toyonaka, Osaka,560-8531, Japan (M. Uchida)
Graduate School of Engineering Science, and Center for Mathematical Modeling and Date Science,Osaka University, 1-3, Machikaneyama, Toyonaka, Osaka, 560-8531, Japan
Key words and phrases. adaptive tests, change point estimation, diffusion processes, discrete observations, test for parameterchange. a r X i v : . [ m a t h . S T ] F e b Y. TONAKI, Y. KAINO, AND M. UCHIDA
Figure 1.
Sample paths of the hyperbolic diffusion model d X t = (cid:16) β + γX t √ X t (cid:17) d t + α d W t whose parameter changes form ( α, β, γ ) = (0 . , ,
3) to (1 . , ,
3) at t = 60 (left), andchanges from ( α, β, γ ) = (1 , ,
3) to (1 , − ,
3) at t = 40 (right). Figure 2.
Sample paths of the hyperbolic diffusion model d X t = (cid:16) β + γX t √ X t (cid:17) d t + α d W t whose parameter changes form ( α, β, γ ) = (1 , ,
3) to (1 . , ,
3) at t = 30 (left), and changesfrom ( α, β, γ ) = (1 , ,
3) to (1 , . ,
3) at t = 50 (right). STIMATION FOR CHANGE POINT OF ERGODIC DIFFUSION PROCESSES 3 (2012) studied the estimating problem of the change point in diffusion parameter. In contrast, Song and Lee(2009), Lee (2011) and Song (2020) considered the detection of the parameter change in diffusion parameter,and Negri and Nishiyama (2017) and Tonaki et al. (2020) considered the detection of the parameter changein diffusion and drift parameters. Specifically, Negri and Nishiyama (2017) proposed simultaneous test forchanges in diffusion and drift parameters, and Tonaki et al. (2020) proposed adaptive tests for those changes.When we consider the change point problem of the diffusion processes, we first need to investigate whetherthere are any change points in the given data. Negri and Nishiyama (2017) considered the following hypothesistesting problem: H θ : θ = ( α, β ) does not change over 0 ≤ t ≤ T v.s. H θ : not H θ .They considered the simultaneous test for changes in diffusion and drift parameters, but even if any changesare detected, we can not determine which parameter changed. In order to solve this problem, Tonaki et al.(2020) considered the adaptive tests for changes in diffusion and drift parameters. Specifically, consider thefollowing steps. First, consider the following hypothesis testing problem. H α : α does not change over 0 ≤ t ≤ T v.s. H α : not H α .If H α is not rejected, then consider the following hypothesis testing problem. H β : β does not change over 0 ≤ t ≤ T v.s. H β : not H β .When we find that there is a change, the next task is to estimate the change point. Since the study of theestimation of the change point for ergodic diffusion processes is still in its infancy, we consider the estimationof the change point for ergodic diffusion processes. With the adaptive detection method, when H α is rejected,we can consider the estimation of the change point of the diffusion parameter α , and when H α is not rejectedand H β is rejected, we can consider the estimation of the change point of the drift parameter β , which bringsus one step closer to our goal.This paper is organized as follows. In Section 2, we state the main results. We propose change pointestimators for diffusion and drift parameters and present rates of convergence and distributional results ofchange point estimators. Section 3 discusses the nuisance parameters of the proposed estimators. Section 4presents the powers of tests proposed by Tonaki et al. (2020). In Sections 5 and 6, we give some examplesand simulation studies. Finally, we provide the proofs in Section 7.2. Main Results
We consider the following two situations:
Situation I. α ∗ changes in 0 ≤ t ≤ T , that is, there exists τ α ∗ ∈ (0 ,
1) such that α ∗ = (cid:40) α ∗ , t ∈ [0 , τ α ∗ T ) ,α ∗ , t ∈ [ τ α ∗ T, T ] , where α ∗ , α ∗ ∈ IntΘ A , α ∗ (cid:54) = α ∗ . Now (1.1) can be expressed as follows. X t = X + (cid:90) t b ( X s , β )d s + (cid:90) t a ( X s , α ∗ )d W s , t ∈ [0 , τ α ∗ T ) X τ α ∗ T + (cid:90) tτ α ∗ T b ( X s , β )d s + (cid:90) tτ α ∗ T a ( X s , α ∗ )d W s . t ∈ [ τ α ∗ T, T ] Situation II. α ∗ does not change and β ∗ changes in 0 ≤ t ≤ T , that is, there exists τ β ∗ ∈ (0 ,
1) such that β ∗ = (cid:40) β ∗ , t ∈ [0 , τ β ∗ T ) ,β ∗ , t ∈ [ τ β ∗ T, T ] , where β ∗ , β ∗ ∈ IntΘ B , β ∗ (cid:54) = β ∗ . Now (1.1) can be expressed as follows. X t = X + (cid:90) t b ( X s , β ∗ )d s + (cid:90) t a ( X s , α ∗ )d W s , t ∈ [0 , τ β ∗ T ) X τ β ∗ T + (cid:90) tτ β ∗ T b ( X s , β ∗ )d s + (cid:90) tτ β ∗ T a ( X s , α ∗ )d W s . t ∈ [ τ β ∗ T, T ]We consider the following two cases:
Case A.
The parameters α ∗ and α ∗ (resp. β ∗ and β ∗ ) depend on n in Situation I (resp. II), Y. TONAKI, Y. KAINO, AND M. UCHIDA
Case B.
The parameters α ∗ and α ∗ (resp. β ∗ and β ∗ ) are fixed and not depend on n in Situation I (resp.II).For matrices c ∈ R d ⊗ R d , we write c ⊗ = cc T , where c T is the transpose of c . Let A ( x, α ) = a ( x, α ) ⊗ and ∆ X i = X t i − X t i − . Let C k,(cid:96) ↑ ( R d × Θ) be the space of all functions f satisfying the following conditions:(i) f is continuously differentiable with respect to x ∈ R d up to order k for all θ ∈ Θ;(ii) f and all its x -derivatives up to order k are (cid:96) times continuously differentiable with respect to θ ∈ Θ;(iii) f and all derivatives are of polynomial growth in x ∈ R d uniformly in θ ∈ Θ, i.e., g is of polynomialgrowth in x ∈ R d uniformly in θ ∈ Θ if, for some
C >
0, we havesup θ ∈ Θ (cid:107) g ( x, θ ) (cid:107) ≤ C (1 + (cid:107) x (cid:107) ) C . We assume the following conditions: [C1]
There exists a constant C such that for any x, y ∈ R d ,sup α ∈ Θ A (cid:107) a ( x, α ) − a ( y, α ) (cid:107) + sup β ∈ Θ B (cid:107) b ( x, β ) − b ( y, β ) (cid:107) ≤ C (cid:107) x − y (cid:107) . [C2] sup t E θ (cid:2) (cid:107) X t (cid:107) k (cid:3) < ∞ for all k ≥ θ ∈ Θ. [C3] inf x,α det ( A ( x, α )) > [C4] a ∈ C , ↑ ( R d × Θ A ) and b ∈ C , ↑ ( R d × Θ B ). [C5] The solution of (1.1) is ergodic with its invariant measure µ θ such that (cid:90) R d (cid:107) x (cid:107) k d µ θ ( x ) < ∞ for all k ≥ θ ∈ Θ , and (cid:90) R d f ( x )d µ θ n ( x ) −→ (cid:90) R d f ( x )d µ θ ( x ) for all measurable function f as θ n −→ θ .2.1. Estimation for the diffusion parameter.
First, we consider Situation I, that is, the estimation forthe diffusion parameter. Write ϑ α = | α ∗ − α ∗ | and letΞ α ( x, α ) = (cid:104) tr (cid:0) A − ∂ α (cid:96) AA − ∂ α (cid:96) A ( x, α ) (cid:1)(cid:105) p(cid:96) ,(cid:96) =1 , Γ α ( x, α , α ) = tr (cid:0) A − ( x, α ) A ( x, α ) − I d (cid:1) − log det A − ( x, α ) A ( x, α ) . Now we additionally assume the following conditions: [C6-I]
There exist estimators ˆ α k = ˆ α k,n ( k = 1 ,
2) such that √ n (ˆ α k − α ∗ k ) = O p (1) . [A1-I] α ∗ and α ∗ depend on n , and ϑ α = ϑ α,n satisfies the following. ϑ α −→ , nϑ α −→ ∞ , hϑ α −→ ∞ , T ϑ α −→ n → ∞ , and ϑ − α ( α ∗ k − α ) = O (1), where α ∈ Int Θ A , [A2-I] For the following three functions and for any r ∈ (1 ,
2) such that nh r −→ ∞ ,max [ n /r ] ≤ k ≤ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k [ nτ α ∗ ]+ k (cid:88) i =[ nτ α ∗ ]+1 f ( X t i − ) − (cid:90) R d f ( x )d µ α ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p −→ . (a) Ξ α ( x, α ),(b) ∂ α Ξ α ( x, α ),(c) ∂ α (cid:16) tr( A − ( x, α ) A ( x, α )) − log det A − ( x, α ) A ( x, α ) (cid:17)(cid:12)(cid:12)(cid:12) α = α . [B1-I] inf x Γ α ( x, α ∗ , α ∗ ) > [B2-I] There exists a constant
C > x,α k (cid:12)(cid:12) ∂ ( α ,α ) Γ α ( x, α , α ) (cid:12)(cid:12) < C , STIMATION FOR CHANGE POINT OF ERGODIC DIFFUSION PROCESSES 5 (b) sup x,α k (cid:12)(cid:12)(cid:12)(cid:104) tr (cid:16)(cid:0) A − ( x, α ) − A − ( x, α ) (cid:1) ∂ α (cid:96) A ( x, α ) (cid:17)(cid:105) p(cid:96) =1 (cid:12)(cid:12)(cid:12) < C ,(c) sup x,θ | Q ( x, θ ) | < C ,where Q ( x, θ ) is the coefficient of h of E θ [( X t i − X t i − ) ⊗ | G ni − ], that is, E θ [( X t i − X t i − ) ⊗ | G ni − ] = hA ( X t i − , α ) + h Q ( X t i − , θ ) + · · · . Here, G ni − = σ (cid:2) { W s } s ≤ t ni (cid:3) . Remark 1
See Section 3 for how to construct the estimators ˆ α k that satisfy [C6-I] . If a ( x, α ) = σ ( x ) c ( α )for σ : R d −→ R d ⊗ R d , c : R p −→ R d ⊗ R d , then [A2-I] , [B1-I] and [B2-I] (a),(b) hold because the functionsof (a)-(c) of [A2-I] , Γ α ( x, α , α ), ∂ ( α ,α ) Γ α ( x, α , α ) and (cid:104) tr (cid:16)(cid:0) A − ( x, α ) − A − ( x, α ) (cid:1) ∂ α (cid:96) A ( x, α ) (cid:17)(cid:105) p(cid:96) =1 do not depend on x . Therefore, the Ornstein-Uhlenbeck process and hyperbolic diffusion model, which willbe described later in Section 5, are models that satisfy [A2-I] , and these models are examples in Case A.Hyperbolic diffusion model also satisfies [B2-I] (c), and this model is an example in Case B, see Section 5.5.If the diffusion coefficient is a ( x, α ) = α , then [A1-I] and [A2-I] can be reduced to the following condition: [A1’-I] α ∗ and α ∗ depend on n , and ϑ α = ϑ α,n satisfies the following. ϑ α −→ , nϑ α −→ ∞ , T ϑ α −→ n → ∞ , and ϑ − α ( α ∗ k − α ) = O (1), where α ∈ Int Θ A .Therefore, if { X t } t ≥ after the parameter change is stationary, then [A1-I] and [A2-I] can be reduced tothe following condition: [A1”-I] α ∗ and α ∗ depend on n , and ϑ α = ϑ α,n satisfies the following. ϑ α −→ , nϑ α −→ ∞ , hϑ α −→ ∞ as n → ∞ , and ϑ − α ( α ∗ k − α ) = O (1), where α ∈ Int Θ A ,That is, if { X t } t ≥ after the parameter change is stationary and we assume [A1’-I] , then [A1-I] and [A2-I] hold, see Section 5.1.In Situation I, we set F i ( α ) = tr (cid:18) A − ( X t i − , α ) (∆ X i ) ⊗ h (cid:19) + log det A ( X t i − , α ) , Φ n ( τ : α , α ) = [ nτ ] (cid:88) i =1 F i ( α ) + n (cid:88) i =[ nτ ]+1 F i ( α )and propose ˆ τ αn = argmin τ ∈ [0 , Φ n ( τ : ˆ α , ˆ α )as an estimator of τ α ∗ .In Case A, we set for v ∈ R , e α = lim n →∞ ϑ − α ( α ∗ − α ∗ ) , J α = 12 e T α (cid:90) R d Ξ α ( x, α )d µ α ( x ) e α , F ( v ) = − J / α W ( v ) + J α | v | , where W is a two sided standard Wiener process. Theorem 1
Suppose that [C1] - [C5] and [C6-I] hold in Situation I. Then, under [A1-I] and [A2-I] inCase A, nϑ α (ˆ τ αn − τ α ∗ ) d −→ argmin v ∈ R F ( v ) . Y. TONAKI, Y. KAINO, AND M. UCHIDA
Theorem 2
Suppose that [C1] - [C5] and [C6-I] hold in Situation I. Then, under [B1-I] and [B2-I] inCase B, n (ˆ τ αn − τ α ∗ ) = O p (1) . Corollary 1
Suppose that [C1] - [C5] and [C6-I] hold in Situation I. If for (cid:15) ∈ [0 ,
1) there exists δ ∈ (0 , − (cid:15) ) such that nh / ( (cid:15) + δ ) −→
0, then, under [B1-I] , [B2-I] (a) and (b) in Case B, n (cid:15) (ˆ τ αn − τ α ∗ ) = o p (1) . Particularly, for (cid:15) ∈ [0 , ) there exists always δ regardless of h . Remark 2
Although we could not specify the distribution of n (ˆ τ αn − τ α ∗ ) in Case B, we have strong resultsin this case. Actually, in Case B, we see ˆ τ αn − τ α ∗ = O p ( n − ). However, if we choose ϑ α = n − ν , 0 < ν < that satisfies [A1-I] , we have n − ν (ˆ τ αn − τ α ∗ ) = O p (1) in Case A, but n − ν (ˆ τ αn − τ α ∗ ) = o p (1) in Case B.2.2. Estimation for the drift parameter.
Next, we consider Situation II, that is, the estimation for thedrift parameter. Write ϑ β = | β ∗ − β ∗ | and letΞ β ( x, α, β ) = (cid:104) ∂ β (cid:96) b ( x, β ) T A − ( x, α ) ∂ β (cid:96) b ( x, β ) (cid:105) q(cid:96) ,(cid:96) =1 , Γ β ( x, α, β , β ) = tr (cid:2) A − ( x, α )( b ( x, β ) − b ( x, β )) ⊗ (cid:3) . Now we additionally assume the following conditions: [C6-II]
There exist estimators ˆ α = ˆ α n , ˆ β k = ˆ β k,n ( k = 1 ,
2) such that √ n (ˆ α − α ∗ ) = O p (1) , √ T ( ˆ β k − β ∗ k ) = O p (1) . [A1-II] β ∗ and β ∗ depend on n and ϑ β = ϑ β,n satisfies the following. ϑ β −→ , T ϑ β −→ ∞ , T ϑ β −→ n → ∞ , and ϑ − β ( β ∗ k − β ) = O (1), where β ∈ Int Θ B . [A2-II] For the following three functions and for any r ∈ (1 ,
2) such that nh r −→ ∞ ,max [ n /r ] ≤ k ≤ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k [ nτ β ∗ ]+ k (cid:88) i =[ nτ β ∗ ]+1 f ( X t i − ) − (cid:90) R d f ( x )d µ ( α ∗ ,β ) ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p −→ . (a) Ξ β ( x, α ∗ , β ),(b) ∂ β Ξ β ( x, α ∗ , β ),(c) ∂ β (cid:16) tr( A − ( x, α ∗ )( b ( x, β ) − b ( x, β )) ⊗ (cid:17)(cid:12)(cid:12)(cid:12) β = β . [A3-II] There exists an integer m ≥ n m/ − h ( m − / −→ ∞ , h − / ϑ m − β −→ b ∈ C ,m ↑ ( R d × Θ B ). [B1-II] inf x Γ β ( x, α ∗ , β ∗ , β ∗ ) > [B2-II] There exists a constant
C > x,α,β k (cid:12)(cid:12) ∂ ( α,β ,β ) Γ β ( x, α, β , β ) (cid:12)(cid:12) < C ,(b) sup x,α,β k (cid:12)(cid:12)(cid:12)(cid:104) tr (cid:16) A − ( x, α ) ∂ β (cid:96) b ( x, β ) (cid:0) b ( x, β ) − b ( x, β ) (cid:1) T (cid:17)(cid:105) q(cid:96) =1 (cid:12)(cid:12)(cid:12) < C . Remark 3
See Section 3 for how to construct the estimators ˆ β k that satisfy [C6-II] . If { X t } t ≥ after theparameter change is stationary, then [A1-II] and [A2-II] can be reduced to the following condition: [A1’-II] β ∗ and β ∗ depend on n and ϑ β = ϑ β,n satisfies the following. ϑ β −→ , T ϑ β −→ ∞ as n → ∞ , and ϑ − β ( β ∗ k − β ) = O (1), where β ∈ Int Θ B .That is, if { X t } t ≥ after the parameter change is stationary and we assume [A1’-II] , then [A1-II] and [A2-II] hold, see Section 5.1. The hyperbolic diffusion model is one of the models that satisfy [B1-II] and [B2-II] , see Section 5.5. STIMATION FOR CHANGE POINT OF ERGODIC DIFFUSION PROCESSES 7
In Situation II, we set G i ( β | α ) = tr (cid:18) A − ( X t i − , α ) (∆ X i − hb ( X t i − , β )) ⊗ h (cid:19) , Ψ n ( τ : β , β | α ) = [ nτ ] (cid:88) i =1 G i ( β | α ) + n (cid:88) i =[ nτ ]+1 G i ( β | α )and propose ˆ τ βn = argmin τ ∈ [0 , Ψ n ( τ : ˆ β , ˆ β | ˆ α )as an estimator of τ β ∗ .In Case A, we set for v ∈ R , e β = lim n →∞ ϑ − β ( β ∗ − β ∗ ) , J β = e T β (cid:90) R d Ξ β ( x, α ∗ , β )d µ ( α ∗ ,β ) ( x ) e β , G ( v ) = − J / β W ( v ) + J β | v | . Theorem 3
Suppose that [C1] - [C5] and [C6-II] hold in Situation II. Then, under [A1-II] - [A3-II] inCase A, T ϑ β (ˆ τ βn − τ β ∗ ) d −→ argmin v ∈ R G ( v ) . Theorem 4
Suppose that [C1] - [C5] and [C6-II] hold in Situation II. Then, under [B1-II] , [B2-II] inCase B, T (ˆ τ βn − τ β ∗ ) = O p (1) . Remark 4
Although we could not specify the distribution of T (ˆ τ βn − τ β ∗ ) in Case B, we have strong resultsin this case. Actually, in Case B, we see ˆ τ βn − τ β ∗ = O p ( T − ). However, if we choose ϑ β = T − ν , 0 < ν < that satisfies [A1-II] , we have T − ν (ˆ τ βn − τ β ∗ ) = O p (1) in Case A, but T − ν (ˆ τ βn − τ β ∗ ) = o p (1) in Case B.3. Estimation of the nuisance parameters α ∗ k , β ∗ k When the values of the parameters α ∗ k , β ∗ k are unknown, it is necessary to estimate the parameters α ∗ k , β ∗ k for the change point estimation. In this section, we will discuss estimation of nuisance parameters α ∗ k , β ∗ k .First, we need the following information to estimate α ∗ k or β ∗ k : [D1] There exist τ α , τ α ∈ (0 ,
1) such that τ α ∗ ∈ [ τ α , τ α ]. [D2] There exist τ β , τ β ∈ (0 ,
1) such that τ β ∗ ∈ [ τ β , τ β ].If this information is obtained, since there are no change points in the intervals [0 , τ α T ] and [ τ α T, T ], wecan estimate α ∗ from the data of [0 , τ α T ] and α ∗ from the data of [ τ α T, T ]. For example, we can constructestimators that satisfy [C6-I] or [C6-II] by the following procedures. In Situation I, set U (1) n ( α ) = [ nτ α ] (cid:88) i =1 (cid:32) tr (cid:18) A − ( X t i − , α ) (∆ X i ) ⊗ h (cid:19) + log det A ( X t i − , α ) (cid:33) ,U (1) n ( α ) = n (cid:88) i =[ nτ α ]+1 (cid:32) tr (cid:18) A − ( X t i − , α ) (∆ X i ) ⊗ h (cid:19) + log det A ( X t i − , α ) (cid:33) , and let ˆ α = arginf α U (1) n ( α ), and ˆ α = arginf α U (1) n ( α ) as estimators of α ∗ and α ∗ , respectively. Then, wehave √ n (ˆ α − α ∗ ) = O p (1) , √ n (ˆ α − α ∗ ) = O p (1) . Y. TONAKI, Y. KAINO, AND M. UCHIDA
Similarly in Situation II, set U (1) n ( α ) = n (cid:88) i =1 (cid:32) tr (cid:18) A − i − ( α ) (∆ X i ) ⊗ h (cid:19) + log det A i − ( α ) (cid:33) ,U (2) n ( β | α ) = [ nτ β ] (cid:88) i =1 tr (cid:18) A − i − ( α ) (∆ X i − hb i − ( β )) ⊗ h (cid:19) ,U (2) n ( β | α ) = n (cid:88) i =[ nτ β ]+1 tr (cid:18) A − i − ( α ) (∆ X i − hb i − ( β )) ⊗ h (cid:19) , and let ˆ α = arginf α U (1) n ( α ), ˆ β = arginf β U (2) n ( β | ˆ α ), and ˆ β = arginf β U (2) n ( β | ˆ α ) as estimators of α ∗ , β ∗ and β ∗ , respectively. Then, we have √ T ( ˆ β − β ∗ ) = O p (1) , √ T ( ˆ β − β ∗ ) = O p (1) . Thus, if [D1] or [D2] is satisfied, then we can construct estimators that satisfy [C6-I] or [C6-II] . Next,let us discuss how to find τ and τ that satisfy [D1] or [D2] . To find these, we need some method to detectchanges in the diffusion or drift parameters. However, in Situations I and II, the adaptive tests for changes indiffusion and drift parameters proposed by Tonaki et al. (2020) makes it possible. Namely, we can detect achange in the diffusion or drift parameters in the interval [ τ T, τ T ] by the following test statistics T αn ( τ , τ ), T β ,n ( τ , τ ) and T β ,n ( τ , τ ). T αn ( τ , τ ) = 1 (cid:112) d ([ nτ ] − [ nτ ]) max ≤ k ≤ [ nτ ] − [ nτ ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ ]+ k (cid:88) i =[ nτ ]+1 ˆ η i − k [ nτ ] − [ nτ ] [ nτ ] (cid:88) i =[ nτ ]+1 ˆ η i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , ˆ η i = tr (cid:18) A − ( X t i − , ˆ α ) (∆ X i ) ⊗ h (cid:19) , T β ,n ( τ , τ ) = 1 (cid:112) dT ( τ − τ ) max ≤ k ≤ [ nτ ] − [ nτ ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ ]+ k (cid:88) i =[ nτ ]+1 ˆ ξ i − k [ nτ ] − [ nτ ] [ nτ ] (cid:88) i =[ nτ ]+1 ˆ ξ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , ˆ ξ i = 1 T d a − ( X t i − , ˆ α )( X t i − X t i − − hb ( X t i − , ˆ β )) , T β ,n ( τ , τ ) = 1 (cid:112) T ( τ − τ ) max ≤ k ≤ [ nτ ] − [ nτ ] (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) I − / n ( τ , τ ) [ nτ ]+ k (cid:88) i =[ nτ ]+1 ˆ ζ i − k [ nτ ] − [ nτ ] [ nτ ] (cid:88) i =[ nτ ]+1 ˆ ζ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , ˆ ζ i = ∂ β b ( X t i − , ˆ β ) T A − ( X t i − , ˆ α )( X t i − X t i − − hb ( X t i − , ˆ β )) , I n ( τ , τ ) = 1[ nτ ] − [ nτ ] [ nτ ] (cid:88) i =[ nτ ]+1 ∂ β b ( X t i − , ˆ β ) T A − ( X t i − , ˆ α ) ∂ β b ( X t i − , ˆ β ) . Below we describe how to find τ and τ .First, the assumption is that a change is detected in the interval [0 , T ]. U-1)
Choose τ U ∈ (0 ,
1) (e.g., τ U = 3 / , τ U T ].(a) If a change is detected, set τ = τ U and go to L-1).(b) If not detected, go to U-2). U- k ) Choose τ Uk ∈ ( τ Uk − ,
1) (e.g., τ Uk = 1 − − ( k +1) ), and investigate the change point in the interval[0 , τ Uk T ].(a) If a change is detected, set τ = τ Uk and go to L-1).(b) If not detected, go to U-( k + 1)).Assume that we chose τ Uk as τ in U-k). L-1)
Choose τ L ∈ (0 , τ Uk ) (e.g., τ L = 1 / τ Uk − ), and investigate the change point in the interval [ τ L T, T ].(a) If a change is detected, set τ = τ L .(b) If not detected, go to L-2). STIMATION FOR CHANGE POINT OF ERGODIC DIFFUSION PROCESSES 9 L- m ) Choose τ Lm ∈ (0 , τ Lm − ) (e.g., τ Lm = 2 − ( m +1) or τ Lk − m (if k > m )), and investigate the change point inthe interval [ τ Lm T, T ].(a) If a change is detected, set τ = τ Lm .(b) If not detected, go to L-( m + 1)).We may also choose τ and τ at the same time, that is, Choose τ ∈ (0 , /
2) (e.g., τ = 1 / τ T, (1 − τ ) T ].(a) If a change is detected, set τ = τ and τ = 1 − τ .(b) If not detected, go to 2). k ) Choose τ k ∈ (0 , τ k − ) (e.g., τ k = 2 − ( k +1) ), and investigate the change point in the interval [ τ k T, (1 − τ k ) T ].(a) If a change is detected, set τ = τ k and τ = 1 − τ k .(b) If not detected, go to k + 1).We can choose τ and τ in the above manner.4. The powers of tests in Case A
As we saw in Section 3, we need estimators of the nuisance parameters α ∗ k and β ∗ k when estimating thechange point, and have to find τ and τ that satisfy [D1] or [D2] in order to construct the estimators. InSection 3, we tried to find them using the adaptive tests proposed by Tonaki et al. (2020). When using thosetests, it is necessary to check whether the consistency holds. However, the consistency of the tests in CaseB was mentioned, but not in Case A. Thus, in this section, we discuss the consistency of the following tests T αn , T β ,n and T β ,n in Case A. T αn = 1 √ dn max ≤ k ≤ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k (cid:88) i =1 ˆ η i − kn n (cid:88) i =1 ˆ η i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , ˆ η i = tr (cid:18) A − ( X t i − , ˆ α ) (∆ X i ) ⊗ h (cid:19) , T β ,n = 1 √ dT max ≤ k ≤ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k (cid:88) i =1 ˆ ξ i − kn n (cid:88) i =1 ˆ ξ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , ˆ ξ i = 1 T d a − ( X t i − , ˆ α )( X t i − X t i − − hb ( X t i − , ˆ β )) , T β ,n = 1 √ T max ≤ k ≤ n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) I − / n (cid:32) k (cid:88) i =1 ˆ ζ i − kn n (cid:88) i =1 ˆ ζ i (cid:33)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , ˆ ζ i = ∂ β b ( X t i − , ˆ β ) T A − ( X t i − , ˆ α )( X t i − X t i − − hb ( X t i − , ˆ β )) , I n = 1 n n (cid:88) i =1 ∂ β b ( X t i − , ˆ β ) T A − ( X t i − , ˆ α ) ∂ β b ( X t i − , ˆ β ) . First, we consider the power of the test for the diffusion parameter α , that is, the following hypothesistesting problem: H α : α ∗ does not change over 0 ≤ t ≤ T , H α : There exists τ α ∗ ∈ (0 ,
1) such that α ∗ = (cid:40) α ∗ , t ∈ [0 , τ α ∗ T ) ,α ∗ , t ∈ [ τ α ∗ T, T ] , where α ∗ and α ∗ depend on n , and α ∗ (cid:54) = α ∗ . Let ϑ α = | α ∗ − α ∗ | . Now, we assume the following conditions: [E1] Under H α , ϑ α −→ nϑ α −→ ∞ as n → ∞ , and there exist α ∈ Int Θ A and c k ∈ R p such that ϑ − α ( α ∗ k − α ) −→ c k for k = 1 , [E2] Under H α , there exists an estimator ˆ α of α such that ϑ − α (ˆ α − α ) = O p (1). [E3] (cid:90) R d (cid:2) tr (cid:0) A − ∂ α (cid:96) A ( x, α ) (cid:1)(cid:3) (cid:96) d µ α ( x )( c − c ) (cid:54) = 0 under H α . Remark 5 [E2] and [E3] . See Section 5.2 for some discussion of estimator ˆ α that satisfies [E2] . See Section5.3 for models that satisfy [E3] .Let (cid:15) ∈ (0 ,
1) and w k ( (cid:15) ) denote the upper- (cid:15) point of sup ≤ s ≤ (cid:107) B k ( s ) (cid:107) , that is, P ( sup ≤ s ≤ (cid:107) B k ( s ) (cid:107) > w k ( (cid:15) )) = (cid:15) . Proposition 1
Assume [C1] - [C5] , [E1] - [E3] . Then, under H α , P (cid:0) T αn > w ( (cid:15) ) (cid:1) −→ , that is, the test T αn is consistent.In the following, we assume that α ∗ does not change over 0 ≤ t ≤ T . We consider the power of the testfor the drift parameter β , that is, the following hypothesis testing problem: H β : β ∗ does not change over0 ≤ t ≤ T , H β : There exists τ β ∗ ∈ (0 ,
1) such that β ∗ = (cid:40) β ∗ , t ∈ [0 , τ β ∗ T ) ,β ∗ , t ∈ [ τ β ∗ T, T ] , where β ∗ and β ∗ depend on n , and β ∗ (cid:54) = β ∗ . Let ϑ β = | β ∗ − β ∗ | . Now, we assume the following conditions: [F1] Under H β , ϑ β −→ T ϑ β −→ ∞ as n → ∞ , and there exist β ∈ Int Θ B and d k ∈ R q such that ϑ − β ( β ∗ k − β ) −→ d k for k = 1 , [F2] Under H β , there exist ¯ β ∗ with ¯ β ∗ − β = O ( ϑ β ) and estimators ˆ α , ˆ β such that √ n (ˆ α − α ∗ ) = O p (1) , √ T ( ˆ β − ¯ β ∗ ) = O p (1) . [F3] (cid:90) R d T d a − ( x, α ∗ ) ∂ β b ( x, β )d µ ( α ∗ ,β ) ( x )( d − d ) (cid:54) = 0 under H β . Remark 6
See Kessler (1995,1997), Uchida and Yoshida (2011, 2012), Tonaki et al. (2020) for how toconstruct the estimators ˆ α and ˆ β that satisfy [F2] . See Section 5.4 and 5.5 for models that satisfy [F3] . Proposition 2
Assume [C1] - [C5] , [F1] - [F3] . Then, under H β , P (cid:0) T β ,n > w ( (cid:15) ) (cid:1) −→ . We additionally assume the following condition: [F4]
There exists an integer M ≥ h − / ϑ M − β −→ b ∈ C ,M ↑ ( R d × Θ B ). Proposition 3
Assume [C1] - [C5] , [F1] , [F2] and [F4] . Then, under H β , P (cid:0) T β ,n > w q ( (cid:15) ) (cid:1) −→ . Remark 7
For the consistency of the test T β ,n , only [F1] , [F2] and [F4] are sufficient. This is because [F3’] (cid:90) R d ∂ β b ( x, β ) T A − ( x, α ∗ ) ∂ β b ( x, β )d µ β ( x )( d − d ) (cid:54) = 0,which corresponds to [F3] , is always valid since (cid:90) R d ∂ β b ( x, β ) T A − ( x, α ∗ ) ∂ β b ( x, β )d µ β ( x ) is regular and | d − d | = lim n →∞ | ϑ − β ( β ∗ − β ) − ϑ − β ( β ∗ − β ) | = lim n →∞ | ϑ − β ( β ∗ − β ∗ ) | = 1 (cid:54) = 0 . Examples
Sufficient condition of assumptions [A2-I] and [A2-II].
A process { X t } t ≥ with a single changepoint can be expressed as follows. There exists a process { ˜ X t } t ≥ such that X t = ˜ X t ( θ ∗ ), X = x , X t = ˜ X t ( θ ∗ ), X = x , X τT = X τT and X t = (cid:40) X t , ≤ t < τ T,X t , τ T ≤ t ≤ T. If { X t } t ≥ is stationary and θ ∗ −→ θ , then we have, for f ∈ C ↑ ( R d ),max [ n /r ] ≤ k ≤ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k [ nτ ]+ k (cid:88) i =[ nτ ]+1 f ( X t i − ) − (cid:90) R d f ( x )d µ θ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = max [ n /r ] ≤ k ≤ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k [ nτ ]+ k (cid:88) i =[ nτ ]+1 f ( X t i − ) − (cid:90) R d f ( x )d µ θ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) STIMATION FOR CHANGE POINT OF ERGODIC DIFFUSION PROCESSES 11 d = max [ n /r ] ≤ k ≤ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k k (cid:88) i =1 f ( X t i − ) − (cid:90) R d f ( x )d µ θ ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p −→ [A2-I] or [A2-II] holds.5.2. Model that satisfies the assumption [E2].
As an example of the model that satisfies [E2] , weconsider the d -dimensional diffusion process X t = X + (cid:90) t b ( X s , β )d s + (cid:90) t σ ( X s ) δ ( α ∗ )d W s , t ∈ [0 , τ α ∗ T ) X τ α ∗ T + (cid:90) tτ α ∗ T b ( X s , β )d s + (cid:90) tτ α ∗ T σ ( X s ) δ ( α ∗ )d W s , t ∈ [ τ α ∗ T, T ]where σ : R d −→ R d ⊗ R d , δ ( α ) = diag( α , . . . , α d ), α = ( α , . . . , α d ) T , α , . . . , α d >
0. The true value ofthe parameters are α ∗ = α ∗ , ... α ∗ ,d , α ∗ = α ∗ , ... α ∗ ,d , which convergence to α = ( α , , . . . , α ,d ) T . We define U (1) n ( α ) = n (cid:88) i =1 (cid:32) tr (cid:18) A − i − ( α ) (∆ X i ) ⊗ h (cid:19) + log det A i − ( α ) (cid:33) and set the estimator ˆ α = arginf α U (1) n ( α ). Then, we have ϑ − α ( ˆ α − α ) = O p (1) . (5.1) Proof of (5.1) . We see ∂ α j U (1) n ( α ) = 2 α j (cid:32) − α j n (cid:88) i =1 tr (cid:18) [ σ ( X t i − ) T ] − δ (e j ) σ ( X t i − ) − (∆ X i ) ⊗ h (cid:19) + n (cid:33) , where e j = (0 , . . . , , . . . , T . Then, we haveˆ α j = (cid:118)(cid:117)(cid:117)(cid:116) n n (cid:88) i =1 tr (cid:18) [ σ ( X t i − ) T ] − δ (e j ) σ ( X t i − ) − (∆ X i ) ⊗ h (cid:19) On the other hand, we define U (1) n ( α ) = [ nτ α ∗ ] (cid:88) i =1 (cid:32) tr (cid:18) A − i − ( α ) (∆ X i ) ⊗ h (cid:19) + log det A i − ( α ) (cid:33) ,U (1) n ( α ) = n (cid:88) i =[ nτ α ∗ ]+1 (cid:32) tr (cid:18) A − i − ( α ) (∆ X i ) ⊗ h (cid:19) + log det A i − ( α ) (cid:33) and set ˆ α = arginf α U (1) n ( α ), ˆ α = arginf α U (1) n ( α ). In the same way, we haveˆ α ,j = (cid:118)(cid:117)(cid:117)(cid:116) nτ α ∗ ] [ nτ α ∗ ] (cid:88) i =1 tr (cid:18) [ σ ( X t i − ) T ] − δ (e j ) σ ( X t i − ) − (∆ X i ) ⊗ h (cid:19) , ˆ α ,j = (cid:118)(cid:117)(cid:117)(cid:116) n − [ nτ α ∗ ] n (cid:88) i =[ nτ α ∗ ]+1 tr (cid:18) [ σ ( X t i − ) T ] − δ (e j ) σ ( X t i − ) − (∆ X i ) ⊗ h (cid:19) . Noting that ϑ − α ( ˆ α k − α ) = O p (1) andˆ α j = (cid:114) [ nτ α ∗ ] n ˆ α ,j + n − [ nτ α ∗ ] n ˆ α ,j , we obtain (cid:107) ˆ α − α (cid:107) ≤ d (cid:88) j =1 | ˆ α j − α ,j | = d (cid:88) j =1 | ˆ α j − α ,j || ˆ α j + α ,j | = d (cid:88) j =1 | ˆ α j + α ,j | (cid:12)(cid:12)(cid:12)(cid:12) [ nτ α ∗ ] n (ˆ α ,j − α ,j ) + n − [ nτ α ∗ ] n (ˆ α ,j − α ,j ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ d (cid:88) j =1 | ˆ α j + α ,j | ( | ˆ α ,j − α ,j || ˆ α ,j + α ,j | + | ˆ α ,j − α ,j || ˆ α ,j + α ,j | ) ≤ d (cid:88) j =1 ϑ α (cid:18) | ˆ α ,j + α ,j || ˆ α j + α ,j | ϑ − α | ˆ α ,j − α ,j | + | ˆ α ,j + α ,j || ˆ α j + α ,j | ϑ − α | ˆ α ,j − α ,j | (cid:19) . From ˆ α k p −→ α and ˆ α p −→ α , we have | ˆ α k,j + α ,j || ˆ α j + α ,j | = O p (1) and ϑ − α ( ˆ α − α ) = O p (1). (cid:3) From the above, 1-dimensional Ornstein-Uhlenbeck process and hyperbolic diffusion model are modelswhich satisfy [E2] because the diffusion coefficient is a ( x, α ) = α .5.3. Model that satisfies the assumption [E3] and model that does not.
First, as an example of amodel that satisfies [E3] , we consider the d -dimensional diffusion process with the diffusion coefficient a ( x, α ) = σ ( x )diag( α , . . . , α d ) , where σ : R d −→ R d ⊗ R d , α = ( α , . . . , α d ) T , α , . . . , α d >
0. The true value of the parameters are α ∗ = α + ϑ α c and α ∗ = α + ϑ α c , where α = ( α , , . . . , α , ) T , c = ( c , , . . . , c ,d ) T , c = ( c , , . . . , c ,d ) T .Now we have A ( x, α ) = σ ( x )diag( α , . . . , α d ) σ ( x ) T ,A − ( x, α ) = [ σ ( x ) T ] − diag(1 /α , . . . , /α d ) σ ( x ) − ,∂ α j A ( x, α ) = σ ( x )diag(0 , . . . , α j , . . . , σ ( x ) T , tr (cid:0) A − ∂ α j A ( x, α ) (cid:1) = tr (cid:16) [ σ ( x ) T ] − diag(0 , . . . , /α j , . . . , σ ( x ) T (cid:17) = 2 α j , (cid:90) R d (cid:104) tr (cid:0) A − ∂ α (cid:96) A ( x, α ) (cid:1)(cid:105) (cid:96) d µ α ( x )( c − c ) = d (cid:88) j =1 c ,j − c ,j ) α ,j . Therefore if d (cid:88) j =1 c ,j − c ,j α j (cid:54) = 0 , (5.2)then [E3] holds. Especially, we have (5.2) if any of the following cases:(1) c ,j − c ,j ≥ ≤ j ≤ d , and c ,j − c ,j > ≤ j ≤ d .(2) c ,j − c ,j ≤ ≤ j ≤ d , and c ,j − c ,j < ≤ j ≤ d .That is, [E3] holds when only α j (1 ≤ j ≤ d ) changes. From the above, 1-dimensional Ornstein-Uhlenbeckprocess and hyperbolic diffusion model are models which satisfy [E3] because the diffusion coefficient is a ( x, α ) = α .Next, as an example of a model that does not satisfy [E3] , we consider the 2-dimensional diffusion processwith diffusion coefficient a ( x, α ) = (cid:18) α α α (cid:19) , ( α , α > . STIMATION FOR CHANGE POINT OF ERGODIC DIFFUSION PROCESSES 13
The true value of the parameters are α = α , α , α , , α ∗ = α + ϑ α c , c , c , , α ∗ = α + ϑ α c , c , c , . Now we have A ( x, α ) = (cid:18) α α α (cid:19) (cid:18) α α α (cid:19) = (cid:18) α + α α α α α α (cid:19) ,A − ( x, α ) = 1 α α (cid:18) α − α α − α α α + α (cid:19) ,∂ α A ( x, α ) = (cid:18) α
00 0 (cid:19) , ∂ α A ( x, α ) = (cid:18) α α α (cid:19) , ∂ α A ( x, α ) = (cid:18) α α α (cid:19) , tr (cid:0) A − ∂ α A ( x, α ) (cid:1) = 2 α α α α = 2 α , tr (cid:0) A − ∂ α A ( x, α ) (cid:1) = α α − α α α α = 0 , tr (cid:0) A − ∂ α A ( x, α ) (cid:1) = − α α + 2 α α + α α α α = 2 α , (cid:90) R (cid:18) α , , , α , (cid:19) d µ α ( x ) c , − c , c , − c , c , − c , = 2( c , − c , ) α , + 2( c , − c , ) α , . (5.3)Therefore, (5.3) does not equal 0 in the following cases:(1) c ,j − c ,j ≥ j = 1 ,
3, and c ,j − c ,j > j = 1 , c ,j − c ,j ≤ j = 1 ,
3, and c ,j − c ,j < j = 1 , c , − c , = c , − c , = 0, hence when only α changes, [E3] does not hold.5.4. Ornstein-Uhlenbeck process.
We consider the 1-dimensional Ornstein-Uhlenbeck processd X t = − β ( X t − γ )d t + α d W t , X = x . ( α, β > , γ ∈ R ) . In this subsection, we refer to the consistency of tests T αn and T β ,n in Case A.First, let us consider the consistency of the test T αn in Case A. Thus, we consider the following stochasticdifferential equation X t = X − (cid:90) t β ( X s − γ )d s + α ∗ W t , t ∈ [0 , τ α ∗ T ) ,X τ α ∗ T − (cid:90) tτ α ∗ T β ( X s − γ )d s + α ∗ ( W t − W τ α ∗ T ) , t ∈ [ τ α ∗ T, T ] , where α ∗ = α + ϑ α c and α ∗ = α + ϑ α c , which hold [E1] . Further, from Section 5.2 and 5.3, [E2] and [E3] hold. Therefore the test T αn is consistent by Proposition 1.Next, we investigate the consistency of the test T β ,n in Case A. Thus, we consider the following stochasticdifferential equation X t = X − (cid:90) t β ∗ ( X s − γ ∗ )d s + α ∗ W t , t ∈ [0 , τ β ∗ T ) ,X τ β ∗ T − (cid:90) tτ β ∗ T β ∗ ( X s − γ ∗ )d s + α ∗ ( W t − W τ β ∗ T ) , t ∈ [ τ β ∗ T, T ] , where β = (cid:18) β γ (cid:19) , d k = (cid:18) d k, d k, (cid:19) , β ∗ k = (cid:18) β ∗ k γ ∗ k (cid:19) = β + ϑ β d k , which hold [F1] . Further, (cid:90) R α ∗ ( − x + γ , β )d µ ( β ,γ ) ( x )( d − d ) = (cid:18) , β α ∗ (cid:19) (cid:18) d , − d , d , − d , (cid:19) = β α ∗ ( d , − d , ) . Therefore, if d , − d , (cid:54) = 0, that is, if γ changes and β does not change, then [F3] holds, and the test T β ,n is consistent by Proposition 2. However, when β changes and γ does not change, [F3] does not hold.5.5. Hyperbolic diffusion model.
We consider the hyperbolic diffusion modeld X t = (cid:32) β − γX t (cid:112) X t (cid:33) d t + α d W t , X = x . ( α > , β ∈ R , γ > | β | ) . (5.4)In this subsection, we mention the consistency of the tests T αn and T β ,n in Case A and the fact that thismodel is an example in Case B.Let b ( x, β ) = β − γx √ x and a ( x, α ) = α . By the same discussion as Ornstein-Uhlenbeck process(Section 5.4), the test T αn is consistent in Case A. Next, we investigate the consistency of the test T β ,n inCase A. Thus, we consider the following stochastic differential equation X t = X + (cid:90) t (cid:32) β ∗ − γ ∗ X s (cid:112) X s (cid:33) d s + α ∗ W t , t ∈ [0 , τ β ∗ T ) ,X τ β ∗ T + (cid:90) tτ β ∗ T (cid:32) β ∗ − γ ∗ X s (cid:112) X s (cid:33) d s + α ∗ ( W t − W τ β ∗ T ) , t ∈ [ τ β ∗ T, T ] , where β = (cid:18) β γ (cid:19) , d k = (cid:18) d k, d k, (cid:19) , β ∗ k = (cid:18) β ∗ k γ ∗ k (cid:19) = β + ϑ β d k , which hold [F1] . The invariant density of the solution in (5.4) is π ( x ) = m ( x ) M , where m ( x ) = exp (cid:18) α (cid:16) βx − γ (cid:112) x (cid:17)(cid:19) , M = (cid:90) R m ( x )d x. Now, we have (cid:90) R ∂ β b ( x, β ) π ( x )d x = (cid:90) R π ( x )d x = 1 , (cid:90) R ∂ γ b ( x, β ) π ( x )d x = − M (cid:90) R x √ x exp (cid:18) α (cid:16) βx − γ (cid:112) x (cid:17)(cid:19) d x = α γ (cid:18) M (cid:90) R α (cid:18) β − γx √ x (cid:19) exp (cid:18) α (cid:16) βx − γ (cid:112) x (cid:17)(cid:19) d x − βα (cid:19) and (cid:90) R α (cid:18) β − γx √ x (cid:19) exp (cid:18) α (cid:16) βx − γ (cid:112) x (cid:17)(cid:19) d x = (cid:20) exp (cid:18) α (cid:16) βx − γ (cid:112) x (cid:17)(cid:19)(cid:21) x = ∞ x = −∞ = (cid:20) exp (cid:18) − γα (cid:112) x (cid:18) − βγ x √ x (cid:19)(cid:19)(cid:21) x = ∞ x = −∞ = 0 . Hence, we have (cid:90) R ∂ γ b ( x, β ) π ( x )d x = − βγ . STIMATION FOR CHANGE POINT OF ERGODIC DIFFUSION PROCESSES 15
From the above, we obtain (cid:90) R α ∗ (cid:18) , − β γ (cid:19) d µ ( β ,γ ) ( x )( d − d ) = 1 α ∗ (cid:18) , − β γ (cid:19) (cid:18) d , − d , d , − d , (cid:19) = 1 α ∗ (cid:18) ( d , − d , ) − β γ ( d , − d , ) (cid:19) . (5.5)In the following cases, [F3] holds because equation (5.5) does not equal 0:(1) β changes and γ does not change,(2) β (cid:54) = 0, γ changes and β does not change,(3) β > d , − d , < >
0) and d , − d , > < β < d , − d , < >
0) and d , − d , < > [B1-I] , [B2-I] , [B1-II] and [B2-II] . Note that it was mentioned in Remark 1 that [B1-I] and [B2-I] (a), (b) hold. Thus,in the following, we verify that [B2-I] (c), [B1-II] and [B2-II] hold. From the proof of Lemma 1 of Kessler(1997), we can express Q ( x, θ ) = L θ f ( y | x ) | y = x , where f ( y | x ) = ( y − x ) , L θ f ( y | x ) = b ( y, β ) ∂ y f ( y | x ) + A ( y, α )2 ∂ y f ( y | x ) ,L θ f ( y | x ) = L θ [ L θ f ]( y | x ) . The specific calculation is as follows. L θ f ( y | x ) = 2 b ( y, β )( y − x ) + α ,∂ y [ L θ f ]( y | x ) = 2 (cid:0) ∂ y b ( y, β )( y − x ) + b ( y, β ) (cid:1) ,∂ y [ L θ f ]( y | x ) = 2 (cid:0) ∂ y b ( y, β )( y − x ) + 2 ∂ y b ( y, β ) (cid:1) ,L θ f ( y | x ) = 2 b ( y, β ) (cid:0) ∂ y b ( y, β )( y − x ) + b ( y, β ) (cid:1) + α (cid:0) ∂ y b ( y, β )( y − x ) + 2 ∂ y b ( y, β ) (cid:1) , and Q ( x, θ ) = 2 (cid:0) b ( x, β ) + α ∂ x b ( x, β ) (cid:1) . b ( x, β ) and ∂ x b ( x, β ) are bounded, so we have sup x,θ | Q ( x, θ ) | < C .Thus, [B2-I] (c) holds. Moreover, we haveΓ β ( x, α, β , β ) = 1 α (cid:20)(cid:18) β − γ x √ x (cid:19) − (cid:18) β − γ x √ x (cid:19)(cid:21) = 1 α (cid:18) ( β − β ) − ( γ − γ ) x √ x (cid:19) , where β k = ( β k , γ k ) T . Therefore, since − < x √ x < x ∈ R , we have sup x | Γ β ( x, α ∗ , β ∗ , β ∗ ) | > [B1-II] holds.(1) γ ∗ = γ ∗ ,(2) γ ∗ (cid:54) = γ ∗ and β ∗ − β ∗ < − ( γ ∗ − γ ∗ ),(3) γ ∗ (cid:54) = γ ∗ and β ∗ − β ∗ > γ ∗ − γ ∗ .Furthermore, we see, from ∂ α Γ β ( x, α, β , β ) = − α (cid:18) ( β − β ) − ( γ − γ ) x √ x (cid:19) ,∂ β Γ β ( x, α, β , β ) = 2 α (cid:18) ( β − β ) − ( γ − γ ) x √ x (cid:19) ,∂ γ Γ β ( x, α, β , β ) = − xα √ x (cid:18) ( β − β ) − ( γ − γ ) x √ x (cid:19) ,∂ β Γ β ( x, α, β , β ) = − ∂ β Γ β ( x, α, β , β ), ∂ γ Γ β ( x, α, β , β ) = − ∂ γ Γ β ( x, α, β , β ),1 α ∂ β b ( x, β ) (cid:16) b ( x, β ) − b ( x, β ) (cid:17) = 1 α (cid:18) ( β − β ) − ( γ − γ ) x √ x (cid:19) , α ∂ γ b ( x, β ) (cid:16) b ( x, β ) − b ( x, β ) (cid:17) = − xα √ x (cid:18) ( β − β ) − ( γ − γ ) x √ x (cid:19) and the boundedness of x √ x , thatsup x,α, β k (cid:12)(cid:12) ∂ ( α, β , β ) Γ β ( x, α, β , β ) (cid:12)(cid:12) < C, sup x,α, β k (cid:12)(cid:12)(cid:12)(cid:12) α ∂ β b ( x, β ) (cid:16) b ( x, β ) − b ( x, β ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) < C and [B2-II] holds. 6. Simulations
Case A.
In this subsection, we consider the 1-dimensional Ornstein-Uhlenbeck process:d X t = − β ( X t − γ )d t + α d W t , X = x , where α, β > , γ ∈ R .In order to check the asymptotic behavior of the estimator ˆ τ αn in Case A, we consider the followingstochastic differential equation X t = X − (cid:90) t β ∗ ( X s − γ ∗ )d s + α ∗ W t , t ∈ [0 , τ α ∗ T ) ,X τ α ∗ T − (cid:90) tτ α ∗ T β ∗ ( X s − γ ∗ )d s + α ∗ ( W t − W τ α ∗ T ) , t ∈ [ τ α ∗ T, T ] , where x = 2, β ∗ = 1, γ ∗ = 2 and τ α ∗ = 0 .
5. The number of iteration is 1000. We set that the sample size of thedata { X t i } ni =0 is n = 10 and h n = n − / = 10 − . Note that T = nh n = n / = 10 , nh n = n − / = 10 − , ϑ α = n − . ≈ . nϑ α = n . ≈ . T ϑ α = n − / ≈ . α = 0 . α ∗ = α + n − . and α ∗ = α = 0 .
1. The existence of a change point in the intervals [1 / T, T ] and [0 , / T ] was investigated usingthe method of Tonaki et al. (2020). In all 1000 iterations, the change point was detected. Therefore, weestimated ˆ α and ˆ α with τ α = 1 / τ α = 3 /
4, respectively. The estimates of α ∗ , α ∗ and τ α ∗ are reportedin Table 1. Moreover, one has that for v ∈ R , e α = lim n →∞ ϑ − α ( α ∗ − α ∗ ) = 1 , J α = 12 e T α (cid:90) R d Ξ α ( x, α )d µ α ( x ) e α = 2 α , F ( v ) = − J / α W ( v ) + J α | v | = − (cid:18) J / α W ( v ) − |J α v | (cid:19) d = − (cid:18) W ( J α v ) − |J α v | (cid:19) . It follows from Theorem 1 that nϑ α (ˆ τ αn − τ α ∗ ) d −→ argmin v ∈ R F ( v ) . For v ∈ R , let G ( v ) = W ( v ) − | v | and ˆ η = inf { η ∈ R | G ( η ) = sup v ∈ R G ( v ) } . For the probability densityfunction of the distribution of ˆ η , see Lemma 1.6.3 of Cs¨org¨o and Horv´ath (1997). From Figure 3, we can seethat the distribution of the estimator almost corresponds with the asymptotic distribution in Theorem 1 andthe estimators have good performance. Table 1.
Mean and standard deviation of the estimators under n = 10 , T = 100, h = 10 − , τ α ∗ = 0 . α ∗ ≈ . α ∗ = 0 . α ˆ α ˆ τ αn STIMATION FOR CHANGE POINT OF ERGODIC DIFFUSION PROCESSES 17
N * Thetaa^2 * (tau_hat[, 2] − tau) D en s i t y −0.10 −0.05 0.00 0.05 0.10 −0.10 −0.05 0.00 0.05 0.10 . . . . . . −0.10 −0.05 0.00 0.05 0.10 . . . . . . x v Figure 3.
The figure on the left is the histogram of nϑ α (ˆ τ αn − τ α ∗ ) (black line) and thetheoretical density function (red line). The figure on the right is the empirical distributionfunction of nϑ α (ˆ τ αn − τ α ∗ ) (black line) and the theoretical distribution function (red line).For the simulations of the estimator ˆ τ βn in Case A, we treat the stochastic differential equation as follows. X t = X − (cid:90) t β ∗ ( X s − γ ∗ )d s + α ∗ W t , t ∈ [0 , τ β ∗ T ) ,X τ β ∗ T − (cid:90) tτ β ∗ T β ∗ ( X s − γ ∗ )d s + α ∗ ( W t − W τ β ∗ T ) , t ∈ [ τ β ∗ T, T ] , where x = 5, α ∗ = 0 . β ∗ = 2 . τ β ∗ = 0 .
5. The number of iteration is 1000. We set that the samplesize of the data { X t i } ni =0 is n = 10 and h n = n − / ≈ . × − . Note that T = nh n = n / ≈ . × , nh n = n − / ≈ . ϑ β = n − / ≈ . T ϑ β = n / ≈ . T ϑ β = n − / ≈ . γ ∗ = 5 + ϑ β and γ ∗ = 5. In all iterations, the change point was detected in the intervals [1 / T, T ] and [0 , / T ]. Therefore,we estimated ˆ β and ˆ β with τ β = 1 / τ β = 3 /
4, respectively. The estimates of β ∗ , β ∗ and τ β ∗ arereported in Table 2. Noting that β ∗ = ( β ∗ , γ ∗ ) T , β ∗ = ( β ∗ , γ ∗ ) T , β = ( β ∗ , γ ∗ ) T , µ ( α ∗ , β ) ∼ N (cid:0) γ ∗ , ( α ∗ ) β ∗ (cid:1) and Ξ β ( x, α, β ) = 1 α (cid:18) ( x − γ ) − β ( x − γ ) − β ( x − γ ) β (cid:19) , we obtain that for v ∈ R , e β = lim n →∞ ϑ − β ( β ∗ − β ∗ ) = (cid:18) (cid:19) , J β = e T β (cid:90) R d Ξ β ( x, α ∗ , β )d µ ( α ∗ , β ) ( x ) e β = 1( α ∗ ) (0 , (cid:32) ( α ∗ ) β ∗
00 ( β ∗ ) (cid:33) (cid:18) (cid:19) = (cid:18) β ∗ α ∗ (cid:19) , G ( v ) = − J / β W ( v ) + J β | v | d = − (cid:18) W ( J β v ) − |J β v | (cid:19) . By Theorem 3, we have that
T ϑ β (ˆ τ βn − τ β ∗ ) d −→ argmin v ∈ R G ( v ) . Since Figure 4 shows that the distribution of the estimator is similar to the asymptotic distribution inTheorem 3, the estimator has good behavior.6.2.
Case B.
In this subsection, we consider the hyperbolic diffusion modeld X t = (cid:32) β − γX t (cid:112) X t (cid:33) d t + α d W t , X = x . ( α > , β ∈ R , γ > | β | ) . Table 2.
Mean and standard deviation of the estimators under n = 10 , T ≈ . × , h ≈ . × − , τ β ∗ = 0 . β ∗ = 2 . γ ∗ ≈ . γ ∗ = 5 in Case A.ˆ β ˆ γ ˆ β ˆ γ ˆ τ βn Ter * thetab^2 * (tau_hat[, 2] − tau) D en s i t y −1.0 −0.5 0.0 0.5 1.0 1.5 . . . . . . . . −1.0 −0.5 0.0 0.5 1.0 1.5 . . . . . . −1.0 −0.5 0.0 0.5 1.0 1.5 . . . . . . x v Figure 4.
The figure on the left is the histogram of
T ϑ β (ˆ τ βn − τ β ∗ ) (black line) and thetheoretical density function (red line). The figure on the right is the empirical distributionfunction of T ϑ β (ˆ τ βn − τ β ∗ ) (black line) and the theoretical distribution function (red line).For simulations of the estimator ˆ τ αn in Case B, we study the following stochastic differential equation X t = X + (cid:90) t (cid:32) β ∗ − γ ∗ X s (cid:112) X s (cid:33) d s + α ∗ W t , t ∈ [0 , τ β ∗ T ) ,X τ β ∗ T + (cid:90) tτ β ∗ T (cid:32) β ∗ − γ ∗ X s (cid:112) X s (cid:33) d s + α ∗ ( W t − W τ β ∗ T ) , t ∈ [ τ β ∗ T, T ] , where x = 2, β ∗ = 0, γ ∗ = 1, τ α ∗ = 0 . α ∗ = 1, α ∗ = 2. The number of iteration is 1000. We set thatthe sample size of the data { X t i } ni =0 is n = 10 or 10 and h n = n − / . The existence of a change pointin the intervals [1 / T, T ] and [0 , / T ] was investigated and the change point was detected. Therefore, weestimated ˆ α and ˆ α with τ α = 1 / τ α = 3 /
4, respectively. The estimates of α ∗ , α ∗ and τ α ∗ are reportedin Table 3. By Theorem 2, one has that n (ˆ τ αn − τ α ∗ ) = O p (1) . Since Figure 5 shows that n (ˆ τ αn − τ α ∗ ) does not diverges when increasing from n = 10 to n = 10 , it seemsthat n (ˆ τ αn − τ α ∗ ) is O p (1) in this example. Table 3.
Mean and standard deviation of the estimators under τ α ∗ = 0 . α ∗ = 1, α ∗ = 2in Case B. n T h ˆ α ˆ α ˆ τ αn . . × − − STIMATION FOR CHANGE POINT OF ERGODIC DIFFUSION PROCESSES 19
Histogram of n * (tau_hat[, 2] − tau) n * (tau_hat[, 2] − tau) F r equen cy −540 −520 −500 −480 −460 −440 −420 Histogram of n * (tau_hat[, 2] − tau) n * (tau_hat[, 2] − tau) F r equen cy −600 −500 −400 −300 −200 −100 0 Figure 5.
The figure on the left is the histogram of n (ˆ τ αn − τ α ∗ ) with n = 10 . The figureon the right is the histogram of n (ˆ τ αn − τ α ∗ ) with n = 10 .In order to investigate the asymptotic performance of the estimator ˆ τ βn in Case B, we consider the stochasticdifferential equation as follows. X t = X + (cid:90) t (cid:32) β ∗ − γ ∗ X s (cid:112) X s (cid:33) d s + α ∗ W t , t ∈ [0 , τ β ∗ T ) ,X τ β ∗ T + (cid:90) tτ β ∗ T (cid:32) β ∗ − γ ∗ X s (cid:112) X s (cid:33) d s + α ∗ ( W t − W τ β ∗ T ) , t ∈ [ τ β ∗ T, T ] , where x = 0 . α ∗ = 0 . γ ∗ = 1 . τ β ∗ = 0 . β ∗ = 0 . β ∗ = − .
25. The number of iteration is 1000. Weset that the sample size of the data { X t i } ni =0 is n = 10 or 10 and h n = n − / . In all iterations, the changepoint was detected in the intervals [1 / T, T ] and [0 , / T ]. Therefore, we estimated ˆ β and ˆ β with τ β = 1 / τ β = 3 /
4, respectively. The estimates of β ∗ , β ∗ and τ β ∗ are reported in Table 4. It follows from Theorem4 that T (ˆ τ βn − τ β ∗ ) = O p (1) . From Figure 6, T (ˆ τ βn − τ β ∗ ) does not diverge when increasing from n = 10 to n = 10 . In this example, itappears that T (ˆ τ βn − τ β ∗ ) satisfies O p (1). Table 4.
Mean and standard deviation of the estimators under τ β ∗ = 0 . β ∗ = 0 . β ∗ = − . γ ∗ = 1 . n T h ˆ β ˆ γ ˆ β ˆ γ ˆ τ βn . × . × − − . − − . Proofs
Let G ni − = σ (cid:2) { W s } s ≤ t ni (cid:3) , and C , C (cid:48) > f is a function on R d × Θ, f i − ( θ )denotes the value f ( X t i − , θ ). If { u n } is a positive sequence, R denotes a function on R d × R + × Θ for whichthere exists a constant
C > θ ∈ Θ (cid:107) R ( x, u n , θ ) (cid:107) ≤ u n C (1 + (cid:107) x (cid:107) ) C . Histogram of Ter * (tau_hat[, 2] − tau)
Ter * (tau_hat[, 2] − tau) D en s i t y −6 −4 −2 0 2 4 . . . . . . Histogram of Ter * (tau_hat[, 2] − tau)
Ter * (tau_hat[, 2] − tau) D en s i t y −4 −2 0 2 . . . . . . Figure 6.
The figure on the left is the histogram of T (ˆ τ βn − τ β ∗ ) with n = 10 . The figureon the right is the histogram of T (ˆ τ βn − τ β ∗ ) with n = 10 .Let R i − ( u n , θ ) = R ( X t i − , u n , θ ). We set A ⊗ x ⊗ k = d (cid:88) (cid:96) ,...,(cid:96) k =1 A (cid:96) ,...,(cid:96) k x (cid:96) · · · x (cid:96) k , for A ∈ R d ⊗ · · · ⊗ R d (cid:124) (cid:123)(cid:122) (cid:125) k , x ∈ R d . Lemma 1
Let Υ n ( τ : θ , θ ) be a contrast function, and let ˆ θ , ˆ θ be estimators of θ , θ , respectively, and letˆ τ n = argmin τ ∈ [0 , Υ n ( τ : ˆ θ , ˆ θ ) be the estimator of τ ∗ , and let ˆ H n ( v ) = Υ n ( τ ∗ + v/r n : ˆ θ , ˆ θ ) − Υ n ( τ ∗ : ˆ θ , ˆ θ ).If there exist a positive sequence { r n } with r n −→ ∞ and a random field H ( v ) that satisfy the followingconditions, then r n (ˆ τ n − τ ∗ ) d −→ argmin v ∈ R H ( v ).(a) r n (ˆ τ n − τ ∗ ) = O p (1),(b) For all L >
0, ˆ H n ( v ) w −→ H ( v ) in D [ − L, L ] . Proof.
Set v † = argmin v ∈ R H ( v ). For all x ∈ R , P ( r n (ˆ τ n − τ ∗ ) ≤ x ) ≤ P (cid:18) r n (ˆ τ n − τ ∗ ) ≤ x, r n (ˆ τ n − τ ∗ ) ∈ [ − L, L ] , inf v ∈ [ − L,x ] ˆ H n ( v ) > inf v ∈ [ x,L ] ˆ H n ( v ) (cid:19) + P (cid:16) r n (ˆ τ n − τ ∗ ) / ∈ [ − L, L ] (cid:17) + P (cid:18) inf v ∈ [ − L,x ] ˆ H n ( v ) ≤ inf v ∈ [ x,L ] ˆ H n ( v ) (cid:19) . (7.1)If r n (ˆ τ n − τ ∗ ) ∈ [ − L, L ] and inf v ∈ [ − L,x ] ˆ H n ( v ) > inf v ∈ [ x,L ] ˆ H n ( v ), theninf v ∈ [ − L,x ] Υ n (cid:18) τ ∗ + vr n : ˆ θ , ˆ θ (cid:19) > inf v ∈ [ x,L ] Υ n (cid:18) τ ∗ + vr n : ˆ θ , ˆ θ (cid:19) , hence, from τ ∗ + xr n < ˆ τ n ≤ τ ∗ + Lr n , x < r n (ˆ τ n − τ ∗ ) ≤ L and P (cid:18) r n (ˆ τ n − τ ∗ ) ≤ x, r n (ˆ τ n − τ ∗ ) ∈ [ − L, L ] , inf v ∈ [ − L,x ] ˆ H n ( v ) > inf v ∈ [ x,L ] ˆ H n ( v ) (cid:19) ≤ P (cid:16) r n (ˆ τ n − τ ∗ ) ≤ x, x < r n (ˆ τ n − τ ∗ ) ≤ L (cid:17) = 0 . From (b), as n → ∞ , (7.1) is evaluated as follows.lim n →∞ P (cid:16) r n (ˆ τ n − τ ∗ ) ≤ x (cid:17) ≤ lim n →∞ P (cid:16) r n (ˆ τ n − τ ∗ ) / ∈ [ − L, L ] (cid:17) + lim n →∞ P (cid:18) inf v ∈ [ − L,x ] ˆ H n ( v ) ≤ inf v ∈ [ x,L ] ˆ H n ( v ) (cid:19) ≤ sup n ∈ N P (cid:16) r n (ˆ τ n − τ ∗ ) / ∈ [ − L, L ] (cid:17) + P (cid:18) inf v ∈ [ − L,x ] H ( v ) ≤ inf v ∈ [ x,L ] H ( v ) (cid:19) . (7.2) STIMATION FOR CHANGE POINT OF ERGODIC DIFFUSION PROCESSES 21
Now we have P (cid:18) inf v ∈ [ − L,x ] H ( v ) ≤ inf v ∈ [ x,L ] H ( v ) (cid:19) ≤ P (cid:18) inf v ∈ [ − L,x ] H ( v ) ≤ inf v ∈ [ x,L ] H ( v ) , v † ∈ [ − L, L ] , v † > x (cid:19) + P ( v † / ∈ [ − L, L ]) + P ( v † ≤ x ) . Because if inf v ∈ [ − L,x ] H ( v ) ≤ inf v ∈ [ x,L ] H ( v ) and v † ∈ [ − L, L ], then − L ≤ v † ≤ x , we have P (cid:18) inf v ∈ [ − L,x ] H ( v ) ≤ inf v ∈ [ x,L ] H ( v ) , v † ∈ [ − L, L ] , v † > x (cid:19) ≤ P ( − L ≤ v † ≤ x, v † > x ) = 0 . Therefore from P (cid:18) inf v ∈ [ − L,x ] H ( v ) ≤ inf v ∈ [ x,L ] H ( v ) (cid:19) ≤ P ( v † / ∈ [ − L, L ]) + P ( v † ≤ x )and (a), as L → ∞ , (7.2) is evaluated as follows.lim n →∞ P (cid:16) r n (ˆ τ n − τ ∗ ) ≤ x (cid:17) ≤ lim L →∞ sup n ∈ N P (cid:16) r n (ˆ τ n − τ ∗ ) / ∈ [ − L, L ] (cid:17) + lim L →∞ P ( v † / ∈ [ − L, L ]) + P ( v † ≤ x )= P ( v † ≤ x ) . (7.3)In the same way, we havelim n →∞ P (cid:16) r n (ˆ τ n − τ ∗ ) > x (cid:17) ≤ lim n →∞ P (cid:16) r n (ˆ τ n − τ ∗ ) / ∈ [ − L, L ] (cid:17) + lim n →∞ P (cid:18) inf v ∈ [ − L,x ] ˆ H n ( v ) ≥ inf v ∈ [ x,L ] ˆ H n ( v ) (cid:19) ≤ sup n ∈ N P (cid:16) r n (ˆ τ n − τ ∗ ) / ∈ [ − L, L ] (cid:17) + P (cid:18) inf v ∈ [ − L,x ] H ( v ) ≥ inf v ∈ [ x,L ] H ( v ) (cid:19) ≤ sup n ∈ N P (cid:16) r n (ˆ τ n − τ ∗ ) / ∈ [ − L, L ] (cid:17) + P ( v † > x ) + P ( v † / ∈ [ − L, L ]) . As L → ∞ , lim n →∞ P (cid:16) r n (ˆ τ n − τ ∗ ) > x (cid:17) ≤ P ( v † > x ) , i.e., lim n →∞ P (cid:16) r n (ˆ τ n − τ ∗ ) ≤ x (cid:17) = lim n →∞ (cid:104) − P (cid:16) r n (ˆ τ n − τ ∗ ) > x (cid:17)(cid:105) = 1 − lim n →∞ P (cid:16) r n (ˆ τ n − τ ∗ ) > x (cid:17) ≥ − P ( v † > x )= P ( v † ≤ x ) . (7.4)From (7.3) and (7.4), we obtain lim n →∞ P (cid:16) r n (ˆ τ n − τ ∗ ) ≤ x (cid:17) = P ( v † ≤ x ) and r n (ˆ τ n − τ ∗ ) d −→ v † . (cid:3) In Case A of Situation I, we set F n ( v ) = Φ n (cid:18) τ α ∗ + vnϑ α : α ∗ , α ∗ (cid:19) − Φ n ( τ α ∗ : α ∗ , α ∗ ) , ˆ F n ( v ) = Φ n (cid:18) τ α ∗ + vnϑ α : ˆ α , ˆ α (cid:19) − Φ n ( τ α ∗ : ˆ α , ˆ α ) , D αn ( v ) = ˆ F n ( v ) − F n ( v ) . Lemma 2
Suppose that [C1] - [C5] , [C6-I] , [A1-I] and [A2-I] hold. Then, for all L > v ∈ [ − L,L ] |D αn ( v ) | p −→ n → ∞ . Proof.
We assume that v >
0. Then, we can express D αn ( v ) = [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 (cid:16) F i (ˆ α ) − F i (ˆ α ) (cid:17) − [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 (cid:16) F i ( α ∗ ) − F i ( α ∗ ) (cid:17) = [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 (cid:20)(cid:18) F i ( α ∗ ) + ∂ α F i ( α ∗ )(ˆ α − α ∗ ) + (cid:90) ∂ α F i ( α ∗ + u (ˆ α − α ∗ ))d u ⊗ (ˆ α − α ∗ ) ⊗ (cid:19) − (cid:18) F i ( α ∗ ) + ∂ α F i ( α ∗ )(ˆ α − α ∗ ) + (cid:90) ∂ α F i ( α ∗ + u (ˆ α − α ∗ ))d u ⊗ (ˆ α − α ∗ ) ⊗ (cid:19)(cid:21) − [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 (cid:16) F i ( α ∗ ) − F i ( α ∗ ) (cid:17) = [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 (cid:16) ∂ α F i ( α ∗ )(ˆ α − α ∗ ) + ∂ α F i ( α ∗ )(ˆ α − α ∗ ) (cid:17) + [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 (cid:18)(cid:90) ∂ α F i ( α ∗ + u (ˆ α − α ∗ ))d u ⊗ (ˆ α − α ∗ ) ⊗ − (cid:90) ∂ α F i ( α ∗ + u (ˆ α − α ∗ ))d u ⊗ (ˆ α − α ∗ ) ⊗ (cid:19) . (7.5)Now we see sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 (cid:90) ∂ α F i ( α ∗ k + u (ˆ α k − α ∗ k ))d u ⊗ (ˆ α k − α ∗ k ) ⊗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ [ nτ α ∗ + L/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∂ α F i ( α ∗ k + u (ˆ α k − α ∗ k ))d u (cid:12)(cid:12)(cid:12)(cid:12) | ˆ α k − α ∗ k | ≤ n [ nτ α ∗ + L/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 sup α ∈ Θ A (cid:12)(cid:12) ∂ α F i ( α ) (cid:12)(cid:12) (cid:0) √ n | ˆ α k − α ∗ k | (cid:1) = O p (cid:18) nϑ α (cid:19) = o p (1) (7.6)and [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 ∂ α F i ( α ∗ k )(ˆ α k − α ∗ k )= [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 (cid:16) ∂ α F i ( α ∗ k ) − E α ∗ [ ∂ α F i ( α ∗ k ) | G ni − ] (cid:17) (ˆ α k − α ∗ k ) + [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 E α ∗ [ ∂ α F i ( α ∗ k ) | G ni − ](ˆ α k − α ∗ k ) . (7.7)By Theorem 2.11 of Hall and Heyde (1980), we have E α ∗ n sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 (cid:0) ∂ α F i ( α ∗ k ) − E α ∗ [ ∂ α F i ( α ∗ k ) | G ni − ] (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cn [ nτ α ∗ + L/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 E α ∗ (cid:104) E α ∗ (cid:104)(cid:12)(cid:12) ∂ α F i ( α ∗ k ) − E α ∗ [ ∂ α F i ( α ∗ k ) | G ni − ] (cid:12)(cid:12) (cid:12)(cid:12)(cid:12) G ni − (cid:105)(cid:105) STIMATION FOR CHANGE POINT OF ERGODIC DIFFUSION PROCESSES 23 ≤ C (cid:48) nϑ α −→ √ n sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 (cid:0) ∂ α F i ( α ∗ k ) − E α ∗ [ ∂ α F i ( α ∗ k ) | G ni − ] (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = o p (1) . (7.8)Moreover, we seesup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 E α ∗ [ ∂ α F i ( α ∗ k ) | G ni − ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | ˆ α k − α ∗ k | = sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 (cid:18)(cid:104) tr (cid:16) A − i − ∂ α (cid:96) A i − A − i − ( α ∗ k )( A i − ( α ∗ ) − A i − ( α ∗ k )) (cid:17)(cid:105) (cid:96) + R i − ( h, θ ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | ˆ α k − α ∗ k | = sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 (cid:16) Ξ αi − ( α ∗ k )( α ∗ − α ∗ k )+ (cid:90) (1 − u ) (cid:0) tr (cid:2) A − i − ∂ α (cid:96) A i − A − i − ( α ∗ k ) ∂ α (cid:96) ∂ α (cid:96) A i − ( α ∗ k + u ( α ∗ − α ∗ k )) (cid:3)(cid:1) (cid:96) ,(cid:96) ,(cid:96) d u ⊗ ( α ∗ − α ∗ k ) ⊗ + R i − ( h, θ ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | ˆ α k − α ∗ k | = sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 (cid:18) Ξ αi − ( α )( α ∗ − α ∗ k ) + (cid:90) ∂ α Ξ αi − ( α + u ( α ∗ k − α ))d u ⊗ ( α ∗ − α ∗ k ) ⊗ ( α ∗ k − α )+ (cid:90) (1 − u ) (cid:0) tr (cid:2) A − i − ∂ α (cid:96) A i − A − i − ( α ∗ k ) ∂ α (cid:96) ∂ α (cid:96) A i − ( α ∗ k + u ( α ∗ − α ∗ k )) (cid:3)(cid:1) (cid:96) ,(cid:96) ,(cid:96) d u ⊗ ( α ∗ − α ∗ k ) ⊗ + R i − ( h, θ ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | ˆ α k − α ∗ k |≤ ϑ α √ n sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 Ξ αi − ( α ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϑ − α | α ∗ − α ∗ k |√ n | ˆ α k − α ∗ k | + ϑ α √ n [ nτ α ∗ + L/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 R i − (1 , θ ) ϑ − α | α ∗ − α ∗ k | ϑ − α | α ∗ − α |√ n | ˆ α k − α ∗ k | + ϑ α √ n [ nτ α ∗ + L/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 R i − (1 , θ )( ϑ − α | α ∗ − α ∗ k | ) √ n | ˆ α k − α ∗ k | + 1 √ n [ nτ α ∗ + L/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 R i − ( h, θ ) √ n | ˆ α k − α ∗ k | = O p (cid:18) √ nϑ α (cid:19) + O p (cid:18) √ n (cid:19) + O p (cid:18) h √ nϑ α (cid:19) = o p (1) . (7.9)Therefore, from (7.5)-(7.9), we have sup v ∈ [0 ,L ] |D αn ( v ) | p −→ . By the similar proof, we see sup v ∈ [ − L, |D αn ( v ) | p −→ (cid:3) Lemma 3
Suppose that [C1] - [C5] , [C6-I] , [A1-I] and [A2-I] hold. Then, for all L > F n ( v ) w −→ F ( v ) in D [ − L, L ] as n → ∞ . Proof.
We consider v >
0. We have F n ( v ) = [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 (cid:16) F i ( α ∗ ) − F i ( α ∗ ) (cid:17) = [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 (cid:18) ∂ α F i ( α ∗ )( α ∗ − α ∗ ) + 12 ∂ α F i ( α ∗ ) ⊗ ( α ∗ − α ∗ ) ⊗ + (cid:90) (1 − u ) ∂ α F i ( α ∗ + u ( α ∗ − α ∗ ))d u ⊗ ( α ∗ − α ∗ ) ⊗ (cid:19) = [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 (cid:18) ∂ α F i ( α ∗ )( α ∗ − α ∗ ) + 12 ∂ α F i ( α ∗ ) ⊗ ( α ∗ − α ∗ ) ⊗ (cid:19) + ¯ o p (1)=: F ,n ( v ) + F ,n ( v ) + ¯ o p (1) , where Y n ( v ) = ¯ o p (1) denotes sup v ∈ [0 ,L ] | Y n ( v ) | = o p (1). Now, we see F ,n ( v ) = [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 (cid:16) ∂ α F i ( α ∗ ) − E α ∗ [ ∂ α F i ( α ∗ ) | G ni − ] (cid:17) ( α ∗ − α ∗ ) + [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 E α ∗ [ ∂ α F i ( α ∗ ) | G ni − ]( α ∗ − α ∗ ) , (7.10)sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 E α ∗ [ ∂ α F i ( α ∗ ) | G ni − ]( α ∗ − α ∗ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ϑ α [ nτ α ∗ + L/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 R i − ( h, θ ) = O p (cid:18) hϑ α (cid:19) = o p (1) , (7.11) [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 E α ∗ (cid:34)(cid:18)(cid:16) ∂ α F i ( α ∗ ) − E α ∗ [ ∂ α F i ( α ∗ ) | G ni − ] (cid:17) ( α ∗ − α ∗ ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G ni − (cid:35) = ( α ∗ − α ∗ ) T [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 E α ∗ (cid:20)(cid:16) ∂ α F i ( α ∗ ) − E α ∗ [ ∂ α F i ( α ∗ ) | G ni − ] (cid:17) T (cid:16) ∂ α F i ( α ∗ ) − E α ∗ [ ∂ α F i ( α ∗ ) | G ni − ] (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) G ni − (cid:21) ( α ∗ − α ∗ )= ( α ∗ − α ∗ ) T [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 (cid:0) αi − ( α ∗ ) + R i − ( h, θ ) (cid:1) ( α ∗ − α ∗ ) p −→ e T α (cid:90) R d Ξ α ( x, α )d µ α ( x ) e α v = 4 J α v (7.12)and [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 E α ∗ (cid:34)(cid:18)(cid:16) ∂ α F i ( α ∗ ) − E α ∗ [ ∂ α F i ( α ∗ ) | G ni − ] (cid:17) ( α ∗ − α ∗ ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) G ni − (cid:35) = [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 ϑ α R i − (1 , θ ) p −→ . (7.13)According to Corollary 3.8 of McLeish (1974), we obtain, from (7.12) and (7.13), [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 (cid:16) ∂ α F i ( α ∗ ) − E α ∗ [ ∂ α F i ( α ∗ ) | G ni − ] (cid:17) ( α ∗ − α ∗ ) w −→ − J / α W ( v ) in D [0 , L ] . (7.14)Further, from (7.10), (7.11) and (7.14), we have F ,n ( v ) w −→ − J / α W ( v ) in D [0 , L ]. STIMATION FOR CHANGE POINT OF ERGODIC DIFFUSION PROCESSES 25
Besides, we see sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 ∂ α F i ( α ∗ ) ⊗ ( α ∗ − α ∗ ) ⊗ − J α v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 (cid:16) ∂ α F i ( α ∗ ) − E α ∗ (cid:2) ∂ α F i ( α ∗ ) (cid:12)(cid:12) G ni − (cid:3)(cid:17) ⊗ ( α ∗ − α ∗ ) ⊗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 E α ∗ (cid:2) ∂ α F i ( α ∗ ) (cid:12)(cid:12) G ni − (cid:3) ⊗ ( α ∗ − α ∗ ) ⊗ − J α v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (7.15)sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 (cid:16) ∂ α F i ( α ∗ ) − E α ∗ (cid:2) ∂ α F i ( α ∗ ) (cid:12)(cid:12) G ni − (cid:3)(cid:17) ⊗ ( α ∗ − α ∗ ) ⊗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p −→ v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 E α ∗ (cid:2) ∂ α F i ( α ∗ ) (cid:12)(cid:12) G ni − (cid:3) ⊗ ( α ∗ − α ∗ ) ⊗ − J α v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 (cid:0) Ξ αi − ( α ∗ ) + R i − ( h, θ ) (cid:1) ⊗ ( α ∗ − α ∗ ) ⊗ − J α v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 Ξ αi − ( α ) ⊗ ( α ∗ − α ∗ ) ⊗ − J α v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + o p (1) p −→ , (7.17)where (7.16) is obtained by E α ∗ sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 (cid:16) ∂ α F i ( α ∗ ) − E α ∗ (cid:2) ∂ α F i ( α ∗ ) (cid:12)(cid:12) G ni − (cid:3)(cid:17) ⊗ ( α ∗ − α ∗ ) ⊗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cϑ α [ nτ α ∗ + L/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 E α ∗ (cid:104)(cid:12)(cid:12) ∂ α F i ( α ∗ ) − E α ∗ (cid:2) ∂ α F i ( α ∗ ) (cid:12)(cid:12) G ni − (cid:3)(cid:12)(cid:12) (cid:105) ≤ C (cid:48) ϑ α −→ v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 Ξ αi − ( α ) ⊗ ( α ∗ − α ∗ ) ⊗ − J α v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ sup v ∈ [0 ,(cid:15) n ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 Ξ αi − ( α ) ⊗ ( α ∗ − α ∗ ) ⊗ − J α v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + sup v ∈ [ (cid:15) n ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 Ξ αi − ( α ) ⊗ ( α ∗ − α ∗ ) ⊗ − J α v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15) n sup v ∈ [0 ,(cid:15) n ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 Ξ αi − ( α ) ⊗ ( α ∗ − α ∗ ) ⊗ − J α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + L sup v ∈ [ (cid:15) n ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v [ nτ α ∗ + v/ϑ α ] (cid:88) i =[ nτ α ∗ ]+1 Ξ αi − ( α ) ⊗ ( α ∗ − α ∗ ) ⊗ − J α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ O p ( (cid:15) n ) + L max [ n /r ] ≤ k ≤ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k [ nτ α ∗ ]+ k (cid:88) i =[ nτ α ∗ ]+1 Ξ αi − ( α ) ⊗ ( ϑ − α ( α ∗ − α ∗ )) ⊗ − J α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = o p (1) . Here { (cid:15) n } ∞ n =1 is a positive sequence such that (cid:15) n −→ (cid:15) n h/ϑ α −→ ∞ , and r is a constant with r ∈ (1 , nh r −→ ∞ . From (7.15)-(7.17), we have sup v ∈ [0 ,L ] | F ,n ( v ) − J α v | p −→
0. Therefore we obtain F n ( v ) w −→ − J / α W ( v ) + J α v in D [0 , L ] . The argument for v < (cid:3)
Proof of Theorem 1.
Let M ≥
1. We have P (cid:0) nϑ α | ˆ τ αn − τ α ∗ | > M (cid:1) = P (cid:0) nϑ α (ˆ τ αn − τ α ∗ ) > M (cid:1) + P (cid:0) nϑ α ( τ α ∗ − ˆ τ αn ) > M (cid:1) . (7.18)For τ > τ α ∗ , we haveΦ n ( τ : α , α ) − Φ n ( τ α ∗ : α , α ) = [ nτ ] (cid:88) i =1 F i ( α ) + n (cid:88) i =[ nτ ]+1 F i ( α ) − [ nτ α ∗ ] (cid:88) i =1 F i ( α ) − n (cid:88) i =[ nτ α ∗ ]+1 F i ( α )= [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 (cid:16) F i ( α ) − F i ( α ) (cid:17) . (7.19)Now, from F i ( α ) − F i ( α ) = F i ( α ) − F i ( α ) − E α ∗ [ F i ( α ) − F i ( α ) | G ni − ]+ tr (cid:0) A − i − ( α ) A i − ( α ) − I d (cid:1) − log det A − i − ( α ) A i − ( α ) − tr (cid:16)(cid:0) A − i − ( α ) − A − i − ( α ) (cid:1) (cid:0) A i − ( α ) − h − E α ∗ [(∆ X i ) ⊗ | G ni − ] (cid:1)(cid:17) , we obtain Φ n ( τ : α , α ) − Φ n ( τ α ∗ : α , α )= [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 (cid:16) F i ( α ) − F i ( α ) − E α ∗ [ F i ( α ) − F i ( α ) | G ni − ] (cid:17) + [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 (cid:16) tr (cid:0) A − i − ( α ) A i − ( α ) − I d (cid:1) − log det A − i − ( α ) A i − ( α ) (cid:17) − [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 tr (cid:16)(cid:0) A − i − ( α ) − A − i − ( α ) (cid:1) (cid:0) A i − ( α ) − h − E α ∗ [(∆ X i ) ⊗ | G ni − ] (cid:1)(cid:17) =: M αn ( τ : α , α ) + A αn ( τ : α , α ) + (cid:37) αn ( τ : α , α ) . Let D αn,M = { τ ∈ [0 , | nϑ α ( τ − τ α ∗ ) > M } . For all δ >
0, we have P (cid:0) nϑ α (ˆ τ n − τ α ∗ ) > M (cid:1) ≤ P (cid:18) inf τ ∈ D αn,M Φ n ( τ : ˆ α , ˆ α ) ≤ Φ n ( τ α ∗ : ˆ α , ˆ α ) (cid:19) = P (cid:18) inf τ ∈ D αn,M (cid:16) Φ n ( τ : ˆ α , ˆ α ) − Φ n ( τ α ∗ : ˆ α , ˆ α ) (cid:17) ≤ (cid:19) = P (cid:18) inf τ ∈ D αn,M (cid:16) M αn ( τ : ˆ α , ˆ α ) + A αn ( τ : ˆ α , ˆ α ) + (cid:37) αn ( τ : ˆ α , ˆ α ) (cid:17) ≤ (cid:19) STIMATION FOR CHANGE POINT OF ERGODIC DIFFUSION PROCESSES 27 ≤ P (cid:18) inf τ ∈ D αn,M M αn ( τ : ˆ α , ˆ α ) + A αn ( τ : ˆ α , ˆ α ) + (cid:37) αn ( τ : ˆ α , ˆ α ) ϑ α ([ nτ ] − [ nτ α ∗ ]) ≤ (cid:19) ≤ P (cid:18) inf τ ∈ D αn,M M αn ( τ : ˆ α , ˆ α ) ϑ α ([ nτ ] − [ nτ α ∗ ]) ≤ − δ (cid:19) + P (cid:18) inf τ ∈ D αn,M A αn ( τ : ˆ α , ˆ α ) ϑ α ([ nτ ] − [ nτ α ∗ ]) ≤ δ (cid:19) + P (cid:18) inf τ ∈ D αn,M (cid:37) αn ( τ : ˆ α , ˆ α ) ϑ α ([ nτ ] − [ nτ α ∗ ]) ≤ − δ (cid:19) ≤ P (cid:32) sup τ ∈ D αn,M |M αn ( τ : ˆ α , ˆ α ) | ϑ α ([ nτ ] − [ nτ α ∗ ]) ≥ δ (cid:33) + P (cid:18) inf τ ∈ D αn,M A αn ( τ : ˆ α , ˆ α ) ϑ α ([ nτ ] − [ nτ α ∗ ]) ≤ δ (cid:19) + P (cid:32) sup τ ∈ D αn,M | (cid:37) αn ( τ : ˆ α , ˆ α ) | ϑ α ([ nτ ] − [ nτ α ∗ ]) ≥ δ (cid:33) =: P α ,n + P α ,n + P α ,n . [i] Evaluation of P α ,n . Choose (cid:15) > O α be an open neighborhood of α . Because ∂ α F i ( α ) iscontinuous with respect to α ∈ Θ A , we can choose ¯ α ∈ O ˆ α so that M αn ( τ : ˆ α , ˆ α ) = [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 (cid:16) F i (ˆ α ) − F i (ˆ α ) − E α ∗ [ F i ( α ) − F i ( α ) | G ni − ] (cid:12)(cid:12) α k =ˆ α k (cid:17) = [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 (cid:16) ∂ α F i (¯ α ) − E α ∗ [ ∂ α F i ( α ) | G ni − ] (cid:12)(cid:12) α =¯ α (cid:17) (ˆ α − ˆ α ) . If ˆ α k ∈ O α , then |M αn ( τ : ˆ α , ˆ α ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 (cid:16) ∂ α F i (¯ α ) − E α ∗ [ ∂ α F i ( α ) | G ni − ] (cid:12)(cid:12) α =¯ α (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | ˆ α − ˆ α |≤ sup α ∈ Θ A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 (cid:0) ∂ α F i ( α ) − E α ∗ [ ∂ α F i ( α ) | G ni − ] (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | ˆ α − ˆ α | =: sup α ∈ Θ A | M αn ( τ : α ) || ˆ α − ˆ α | . Hence we have P α ,n = P (cid:32) sup τ ∈ D αn,M |M αn ( τ : ˆ α , ˆ α ) | ϑ α ([ nτ ] − [ nτ α ∗ ]) ≥ δ (cid:33) ≤ P (cid:32) sup τ ∈ D αn,M |M αn ( τ : ˆ α , ˆ α ) | ϑ α ([ nτ ] − [ nτ α ∗ ]) ≥ δ, | ˆ α − ˆ α | ≤ ϑ α , ˆ α , ˆ α ∈ O α (cid:33) + P ( | ˆ α − ˆ α | > ϑ α ) + P (ˆ α / ∈ O α ) + P (ˆ α / ∈ O α ) ≤ P (cid:32) sup τ ∈ D αn,M sup α ∈ Θ A | M αn ( τ : α ) | ϑ α ([ nτ ] − [ nτ α ∗ ]) | ˆ α − ˆ α | ≥ δ, | ˆ α − ˆ α | ≤ ϑ α (cid:33) + P ( | ˆ α − ˆ α | > ϑ α ) + P (ˆ α / ∈ O α ) + P (ˆ α / ∈ O α ) ≤ P (cid:32) sup τ ∈ D αn,M α ∈ Θ A | M αn ( τ : α ) | ϑ α ([ nτ ] − [ nτ α ∗ ]) ≥ δ (cid:33) + P ( | ˆ α − ˆ α | > ϑ α ) + P (ˆ α / ∈ O α ) + P (ˆ α / ∈ O α ) . (7.20)By the uniform version on the H´ajek-Renyi inequality in Lemma 2 of Iacus and Yoshida (2012), we obtain P (cid:32) sup τ ∈ D αn,M α ∈ Θ A | M αn ( τ : α ) | ϑ α ([ nτ ] − [ nτ α ∗ ]) ≥ δ (cid:33) = P (cid:32) sup τ ∈ D αn,M α ∈ Θ A | M αn ( τ : α ) | [ nτ ] − [ nτ α ∗ ] ≥ δϑ α (cid:33) ≤ P max j>M/ϑ α − j sup α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ α ∗ ]+ j (cid:88) i =[ nτ α ∗ ]+1 (cid:0) ∂ α F i ( α ) − E [ ∂ α F i ( α ) | G ni − ] (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ δϑ α ≤ (cid:88) j>M/ϑ α − C ( δϑ α j ) ≤ C (cid:48) ( δϑ α ) ϑ α M = C (cid:48) δ M =: γ α ( M ) , (7.21)Noting that if | ˆ α k − α ∗ k | ≤ ϑ α / | ˆ α − ˆ α | ≤ | ˆ α − α ∗ | + | α ∗ − α ∗ | + | ˆ α − α ∗ | ≤ ϑ α , i.e., {| ˆ α − ˆ α | > ϑ α } ⊂ (cid:26) | ˆ α − α ∗ | > ϑ α (cid:27) ∪ (cid:26) | ˆ α − α ∗ | > ϑ α (cid:27) , and for sufficiently large n so that P ( | ˆ α k − α ∗ k | > ϑ α / < (cid:15)/ ϑ − α (ˆ α k − α ∗ k ) = o p (1), we see P ( | ˆ α − ˆ α | > ϑ α ) ≤ P (cid:18) | ˆ α − α ∗ | > ϑ α (cid:19) + P (cid:18) | ˆ α − α ∗ | > ϑ α (cid:19) < (cid:15). (7.22)From [C6-I] and [A2-I] , we have ˆ α k p −→ α as n → ∞ and P (ˆ α k / ∈ O α ) < (cid:15) for large n . Therefore, from(7.20)-(7.22) and this, we have P α ,n ≤ γ α ( M ) + 3 (cid:15) for large n .[ii] Evaluation of P α ,n . If ˆ α k ∈ O α , then we havetr (cid:0) A − i − (ˆ α ) A i − (ˆ α ) − I d (cid:1) − log det A − i − (ˆ α ) A i − (ˆ α )= tr (cid:0) A − i − (ˆ α ) A i − (ˆ α ) − I d (cid:1) − log det A − i − (ˆ α ) A i − (ˆ α )+ ∂ α (cid:16) tr (cid:0) A − i − ( α ) A i − (ˆ α ) (cid:1) − log det A − i − ( α ) A i − (ˆ α ) (cid:17)(cid:12)(cid:12)(cid:12) α =ˆ α (ˆ α − ˆ α )+ 12 ∂ α (cid:16) tr (cid:0) A − i − ( α ) A i − (ˆ α ) (cid:1) − log det A − i − ( α ) A i − (ˆ α ) (cid:17)(cid:12)(cid:12)(cid:12) α =ˆ α ⊗ (ˆ α − ˆ α ) ⊗ + 13! ∂ α (cid:16) tr (cid:0) A − i − ( α ) A i − (ˆ α ) (cid:1) − log det A − i − ( α ) A i − (ˆ α ) (cid:17)(cid:12)(cid:12)(cid:12) α =ˆ α ⊗ (ˆ α − ˆ α ) ⊗ + (cid:90) (1 − u ) ∂ α (cid:16) tr (cid:0) A − i − ( α ) A i − (ˆ α ) (cid:1) − log det A − i − ( α ) A i − (ˆ α ) (cid:17)(cid:12)(cid:12)(cid:12) α =ˆ α + u (ˆ α − ˆ α ) d u ⊗ (ˆ α − ˆ α ) ⊗ = 12 Ξ αi − (ˆ α ) ⊗ (ˆ α − ˆ α ) ⊗ + 13! ∂ α (cid:16) tr (cid:0) A − i − ( α ) A i − (ˆ α ) (cid:1) − log det A − i − ( α ) A i − (ˆ α ) (cid:17)(cid:12)(cid:12)(cid:12) α =ˆ α ⊗ (ˆ α − ˆ α ) ⊗ + (cid:90) (1 − u ) ∂ α (cid:16) tr (cid:16) A − i − ( α ) A i − (ˆ α ) (cid:17) − log det A − i − ( α ) A i − (ˆ α ) (cid:17)(cid:12)(cid:12)(cid:12) α =ˆ α + u (ˆ α − ˆ α ) d u ⊗ (ˆ α − ˆ α ) ⊗ = 12 Ξ αi − ( α ) ⊗ (ˆ α − ˆ α ) ⊗ + 12 ∂ α Ξ αi − ( α ) ⊗ (ˆ α − ˆ α ) ⊗ ⊗ (ˆ α − α )+ 12 (cid:90) (1 − u ) ∂ α Ξ αi − ( α + u (ˆ α − α ))d u ⊗ (ˆ α − ˆ α ) ⊗ ⊗ (ˆ α − α ) ⊗ + 13! ∂ α (cid:16) tr (cid:0) A − i − ( α ) A i − (ˆ α ) (cid:1) − log det A − i − ( α ) A i − (ˆ α ) (cid:17)(cid:12)(cid:12)(cid:12) α =ˆ α ⊗ (ˆ α − ˆ α ) ⊗ + (cid:90) (1 − u ) ∂ α (cid:16) tr (cid:0) A − i − ( α ) A i − (ˆ α ) (cid:1) − log det A − i − ( α ) A i − (ˆ α ) (cid:17)(cid:12)(cid:12)(cid:12) α =ˆ α + u (ˆ α − ˆ α ) d u ⊗ (ˆ α − ˆ α ) ⊗ ≥ (cid:18) λ [Ξ αi − ( α )] + r i − (cid:19) | ˆ α − ˆ α | , STIMATION FOR CHANGE POINT OF ERGODIC DIFFUSION PROCESSES 29 where λ [ N ] denotes the minimum eigenvalue of a symmetric matrix N , and r i − satisfiessup τ ∈ D αn,M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) nτ ] − [ nτ α ∗ ] [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 r i − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = o p (1)from [A1-I] and [A2-I] . Therefore we obtain P α ,n = P (cid:18) inf τ ∈ D αn,M A αn ( τ : ˆ α , ˆ α ) ϑ α ([ nτ ] − [ nτ α ∗ ]) ≤ δ (cid:19) ≤ P (cid:18) inf τ ∈ D αn,M A αn ( τ : ˆ α , ˆ α ) ϑ α ([ nτ ] − [ nτ α ∗ ]) ≤ δ, | ˆ α − ˆ α | ≥ ϑ α , ˆ α k ∈ O α (cid:19) + P (cid:18) | ˆ α − ˆ α | < ϑ α (cid:19) + P (ˆ α / ∈ O α ) + P (ˆ α / ∈ O α ) ≤ P inf τ ∈ D αn,M ϑ α ([ nτ ] − [ nτ α ∗ ]) [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 (cid:18) λ [Ξ αi − ( α )] + r i − (cid:19) | ˆ α − ˆ α | ≤ δ, | ˆ α − ˆ α | ≥ ϑ α + P (cid:18) | ˆ α − ˆ α | < ϑ α (cid:19) + P (ˆ α / ∈ O α ) + P (ˆ α / ∈ O α ) ≤ P inf τ ∈ D αn,M nτ ] − [ nτ α ∗ ] [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 (cid:18) λ [Ξ αi − ( α )] + r i − (cid:19) ≤ δ + P (cid:18) | ˆ α − ˆ α | < ϑ α (cid:19) + P (ˆ α / ∈ O α ) + P (ˆ α / ∈ O α ) . (7.23)According to [A2-I] , if we set δ = 119 (cid:90) R d λ [Ξ α ( x, α )]d µ α ( x ) > , then for large n , P inf τ ∈ D αn,M nτ ] − [ nτ α ∗ ] [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 (cid:18) λ [Ξ αi − ( α )] + r i − (cid:19) ≤ δ ≤ P inf τ ∈ D αn,M nτ ] − [ nτ α ∗ ] [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 λ [Ξ αi − ( α )] ≤ δ + P inf τ ∈ D αn,M nτ ] − [ nτ α ∗ ] [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 r i − ≤ − δ ≤ P inf τ ∈ D αn,M nτ ] − [ nτ α ∗ ] [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 λ [Ξ αi − ( α )] ≤ δ + P sup τ ∈ D αn,M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) nτ ] − [ nτ α ∗ ] [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 r i − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ δ ≤ P inf τ ∈ D αn,M nτ ] − [ nτ α ∗ ] [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 λ [Ξ αi − ( α )] − δ ≤ − δ + (cid:15) ≤ P sup τ ∈ D αn,M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) nτ ] − [ nτ α ∗ ] [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 λ [Ξ αi − ( α )] − δ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ δ + (cid:15) ≤ P sup k>M/ϑ α − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k [ nτ α ∗ ]+ k (cid:88) i =[ nτ α ∗ ]+1 λ [Ξ αi − ( α )] − δ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ δ + (cid:15) ≤ P max [ n /r ] ≤ k ≤ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k [ nτ α ∗ ]+ k (cid:88) i =[ nτ α ∗ ]+1 λ [Ξ αi − ( α )] − δ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ δ + (cid:15) ≤ (cid:15). (7.24) Noting that if | ˆ α k − α ∗ k | ≤ ϑ α / ϑ α = | α ∗ − α ∗ | ≤ | α ∗ − ˆ α | + | ˆ α − ˆ α | + | ˆ α − α ∗ | ≤ | ˆ α − ˆ α | + ϑ α / (cid:26) | ˆ α − ˆ α | < ϑ α (cid:27) ⊂ (cid:26) | ˆ α − α ∗ | > ϑ α (cid:27) ∪ (cid:26) | ˆ α − α ∗ | > ϑ α (cid:27) and for sufficiently large n so that P ( | ˆ α k − α ∗ k | > ϑ α / < (cid:15)/ ϑ − α (ˆ α k − α ∗ k ) = o p (1), we see P (cid:18) | ˆ α − ˆ α | < ϑ α (cid:19) ≤ P (cid:18) | ˆ α − α ∗ | > ϑ α (cid:19) + P (cid:18) | ˆ α − α ∗ | > ϑ α (cid:19) < (cid:15). (7.25)Therefore, from (7.23)-(7.25), we obtain P α ,n ≤ (cid:15) for large n .[iii] Evaluation of P α ,n . We have, for large n ,tr (cid:16)(cid:0) A − i − (ˆ α ) − A − i − (ˆ α ) (cid:1) (cid:0) A i − (ˆ α ) − h − E α ∗ [(∆ X i ) ⊗ | G ni − ] (cid:1)(cid:17) = tr (cid:16)(cid:0) A − i − (ˆ α ) − A − i − (ˆ α ) (cid:1) ( A i − (ˆ α ) − A i − ( α ∗ ) + R i − ( h, θ )) (cid:17) ≤ tr (cid:16)(cid:0) A − i − (ˆ α ) − A − i − (ˆ α ) (cid:1) ( A i − (ˆ α ) − A i − ( α ∗ )) (cid:17) + (cid:12)(cid:12) A − i − (ˆ α ) − A − i − (ˆ α ) (cid:12)(cid:12) | R i − ( h, θ ) |≤ (cid:104) tr (cid:16) ∂ α (cid:96) A − i − (ˆ α )( A i − (ˆ α ) − A i − ( α ∗ )) (cid:17)(cid:105) (cid:96) (ˆ α − ˆ α )+ (cid:90) (1 − u ) (cid:104) tr (cid:16) ∂ α (cid:96) ∂ α (cid:96) A − i − (ˆ α + u (ˆ α − ˆ α ))( A i − (ˆ α ) − A i − ( α ∗ )) (cid:17)(cid:105) (cid:96) ,(cid:96) d u ⊗ (ˆ α − ˆ α ) ⊗ + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∂ α A − i − (ˆ α + u (ˆ α − ˆ α ))d u (cid:12)(cid:12)(cid:12)(cid:12) | R i − ( h, θ ) || ˆ α − ˆ α | = (cid:2) tr (cid:0) ∂ α (cid:96) A − i − (ˆ α ) ∂ α (cid:96) A i − ( α ∗ ) (cid:1)(cid:3) (cid:96) ,(cid:96) ⊗ (ˆ α − ˆ α ) ⊗ (ˆ α − α ∗ )+ (cid:90) (1 − u ) (cid:104) tr (cid:16) ∂ α (cid:96) A − i − (ˆ α ) ∂ α (cid:96) ∂ α (cid:96) A i − ( α ∗ + u (ˆ α − α ∗ )) (cid:17)(cid:105) (cid:96) ,(cid:96) ,(cid:96) d u ⊗ (ˆ α − ˆ α ) ⊗ (ˆ α − α ∗ ) ⊗ + (cid:90) (1 − v ) (cid:90) (1 − u ) (cid:104) tr (cid:16) ∂ α (cid:96) ∂ α (cid:96) A − i − (ˆ α + u (ˆ α − ˆ α )) ∂ α (cid:96) A i − ( α ∗ + v (ˆ α − α ∗ )) (cid:17)(cid:105) (cid:96) ,(cid:96) ,(cid:96) d u d v ⊗ (ˆ α − ˆ α ) ⊗ ⊗ (ˆ α − α ∗ )+ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∂ α A − i − (ˆ α + u (ˆ α − ˆ α ))d u (cid:12)(cid:12)(cid:12)(cid:12) | R i − ( h, θ ) || ˆ α − ˆ α | = Ξ αi − ( α ∗ ) ⊗ (ˆ α − ˆ α ) ⊗ (ˆ α − α ∗ )+ (cid:90) (cid:104) tr (cid:16) ∂ α (cid:96) ∂ α (cid:96) A − i − ( α ∗ + u (ˆ α − α ∗ )) ∂ α (cid:96) A i − ( α ∗ ) (cid:17)(cid:105) (cid:96) ,(cid:96) ,(cid:96) d u ⊗ (ˆ α − ˆ α ) ⊗ (ˆ α − α ∗ ) ⊗ + (cid:90) (1 − u ) (cid:104) tr (cid:16) ∂ α (cid:96) A − i − (ˆ α ) ∂ α (cid:96) ∂ α (cid:96) A i − ( α ∗ + u (ˆ α − α ∗ )) (cid:17)(cid:105) (cid:96) ,(cid:96) ,(cid:96) d u ⊗ (ˆ α − ˆ α ) ⊗ (ˆ α − α ∗ ) ⊗ + (cid:90) (1 − v ) (cid:90) (1 − u ) (cid:104) tr (cid:16) ∂ α (cid:96) ∂ α (cid:96) A − i − (ˆ α + u (ˆ α − ˆ α )) ∂ α (cid:96) A i − ( α ∗ + v (ˆ α − α ∗ )) (cid:17)(cid:105) (cid:96) ,(cid:96) ,(cid:96) d u d v ⊗ (ˆ α − ˆ α ) ⊗ ⊗ (ˆ α − α ∗ )+ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∂ α A − i − (ˆ α + u (ˆ α − ˆ α ))d u (cid:12)(cid:12)(cid:12)(cid:12) | R i − ( h, θ ) || ˆ α − ˆ α | = Ξ αi − ( α ) ⊗ (ˆ α − ˆ α ) ⊗ (ˆ α − α ∗ ) + (cid:90) ∂ α Ξ αi − ( α + u ( α ∗ − α ))d u ⊗ (ˆ α − ˆ α ) ⊗ (ˆ α − α ∗ ) ⊗ ( α ∗ − α )+ (cid:90) (cid:104) tr (cid:16) ∂ α (cid:96) ∂ α (cid:96) A − i − ( α ∗ + u (ˆ α − α ∗ )) ∂ α (cid:96) A i − ( α ∗ ) (cid:17)(cid:105) (cid:96) ,(cid:96) ,(cid:96) d u ⊗ (ˆ α − ˆ α ) ⊗ (ˆ α − α ∗ ) ⊗ + (cid:90) (1 − u ) (cid:104) tr (cid:16) ∂ α (cid:96) A − i − (ˆ α ) ∂ α (cid:96) ∂ α (cid:96) A i − ( α ∗ + u (ˆ α − α ∗ )) (cid:17)(cid:105) (cid:96) ,(cid:96) ,(cid:96) d u ⊗ (ˆ α − ˆ α ) ⊗ (ˆ α − α ∗ ) ⊗ + (cid:90) (1 − v ) (cid:90) (1 − u ) (cid:104) tr (cid:16) ∂ α (cid:96) ∂ α (cid:96) A − i − (ˆ α + u (ˆ α − ˆ α )) ∂ α (cid:96) A i − ( α ∗ + v (ˆ α − α ∗ )) (cid:17)(cid:105) (cid:96) ,(cid:96) ,(cid:96) d u d v ⊗ (ˆ α − ˆ α ) ⊗ ⊗ (ˆ α − α ∗ )+ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∂ α A − i − (ˆ α + u (ˆ α − ˆ α ))d u (cid:12)(cid:12)(cid:12)(cid:12) | R i − ( h, θ ) || ˆ α − ˆ α |≤ Ξ αi − ( α ) ⊗ (ˆ α − ˆ α ) ⊗ (ˆ α − α ∗ )+ ϑ α √ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∂ α Ξ αi − ( α + u ( α ∗ − α ))d u (cid:12)(cid:12)(cid:12)(cid:12) ϑ − α | ˆ α − ˆ α |√ n | ˆ α − α ∗ | ϑ − α | α ∗ − α | + ϑ α n (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) (cid:104) tr (cid:16) ∂ α (cid:96) ∂ α (cid:96) A − i − ( α ∗ + u (ˆ α − α ∗ )) ∂ α (cid:96) A i − ( α ∗ ) (cid:17)(cid:105) (cid:96) ,(cid:96) ,(cid:96) d u (cid:12)(cid:12)(cid:12)(cid:12) ϑ − α | ˆ α − ˆ α | (cid:0) √ n | ˆ α − α ∗ | (cid:1) + ϑ α n (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) (1 − u ) (cid:104) tr (cid:16) ∂ α (cid:96) A − i − (ˆ α ) ∂ α (cid:96) ∂ α (cid:96) A i − ( α ∗ + u (ˆ α − α ∗ )) (cid:17)(cid:105) (cid:96) ,(cid:96) ,(cid:96) d u (cid:12)(cid:12)(cid:12)(cid:12) ϑ − α | ˆ α − ˆ α | (cid:0) √ n | ˆ α − α ∗ | (cid:1) + ϑ α √ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) (1 − v ) (cid:90) (1 − u ) (cid:104) tr (cid:16) ∂ α (cid:96) ∂ α (cid:96) A − i − (ˆ α + u (ˆ α − ˆ α )) ∂ α (cid:96) A i − ( α ∗ + v (ˆ α − α ∗ )) (cid:17)(cid:105) (cid:96) ,(cid:96) ,(cid:96) d u d v (cid:12)(cid:12)(cid:12)(cid:12)(cid:0) ϑ − α | ˆ α − ˆ α | (cid:1) √ n | ˆ α − α ∗ | + hϑ α (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∂ α A − i − (ˆ α + u (ˆ α − ˆ α ))d u (cid:12)(cid:12)(cid:12)(cid:12) R i − (1 , θ ) ϑ − α | ˆ α − ˆ α |≤ ϑ α √ n Ξ αi − ( α ) ⊗ ϑ − α (ˆ α − ˆ α ) ⊗ √ n (ˆ α − α ∗ ) + (cid:18) ϑ α √ n + ϑ α n + hϑ α (cid:19) R i − (1 , θ )and sup τ ∈ D αn,M | (cid:37) αn ( τ : ˆ α , ˆ α ) | ϑ α ([ nτ ] − [ nτ α ∗ ]) ≤ √ nϑ α sup τ ∈ D αn,M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) nτ ] − [ nτ α ∗ ] [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 Ξ αi − ( α ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϑ − α | ˆ α − ˆ α |√ n | ˆ α − α ∗ | + (cid:18) ϑ α √ n + ϑ α n + hϑ α (cid:19) sup τ ∈ D αn,M ϑ α ([ nτ ] − [ nτ α ∗ ]) [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 R i − (1 , θ ) ≤ O p (cid:18) √ nϑ α (cid:19) + O p ( √ nϑ α ) + O p ( ϑ α ) + O p ( T ϑ α ) = o p (1) . Hence, we see P α ,n ≤ P (cid:32) sup τ ∈ D αn,M | (cid:37) αn ( τ : ˆ α , ˆ α ) | ϑ α ([ nτ ] − [ nτ α ∗ ]) ≥ δ, ˆ α , ˆ α ∈ O α (cid:33) + P (ˆ α / ∈ O α ) + P (ˆ α / ∈ O α ) ≤ (cid:15) for large n .[iv] From the evaluations in Steps [i]-[iii], we havelim n →∞ P ( nϑ α (ˆ τ αn − τ α ∗ ) > M ) ≤ γ α ( M ) + 11 (cid:15) for any M ≥ (cid:15) >
0. Thereforelim M →∞ lim n →∞ P ( nϑ α (ˆ τ αn − τ α ∗ ) > M ) ≤ (cid:15). (7.26)Since, for τ < τ α ∗ ,Φ n ( τ : α , α ) − Φ n ( τ α ∗ : α , α )= [ nτ α ∗ ] (cid:88) i =[ nτ ]+1 (cid:16) F i ( α ) − F i ( α ) − E α ∗ [ F i ( α ) − F i ( α ) | G ni − ] (cid:17) + [ nτ α ∗ ] (cid:88) i =[ nτ ]+1 (cid:16) tr (cid:0) A − i − ( α ) A i − ( α ) − I d (cid:1) − log det A − i − ( α ) A i − ( α ) (cid:17) − [ nτ α ∗ ] (cid:88) i =[ nτ ]+1 tr (cid:16)(cid:0) A − i − ( α ) − A − i − ( α ) (cid:1)(cid:0) A i − ( α ) − h − E α ∗ [(∆ X i ) ⊗ | G ni − ] (cid:1)(cid:17) , we obtain, in the same way as above,lim M →∞ lim n →∞ P ( nϑ α ( τ α ∗ − ˆ τ αn ) > M ) ≤ (cid:15). (7.27)We see, from (7.18), (7.26) and (7.27), lim M →∞ lim n →∞ P ( nϑ α | ˆ τ αn − τ α ∗ | > M ) ≤ (cid:15) , which shows nϑ α (ˆ τ αn − τ α ∗ ) = O p (1) . (7.28)From Lemmas 1-3 and (7.28), we obtain nϑ α (ˆ τ αn − τ α ∗ ) d −→ argmin v ∈ R F ( v ) . This completes the proof of Theorem 1. (cid:3)
Proof of Theorem 2.
As with the proof of Theorem 1, it is sufficient to showlim M →∞ lim n →∞ P ( n (ˆ τ αn − τ α ∗ ) > M ) = 0 . Let D αn,M = { τ ∈ [0 , | n ( τ − τ α ∗ ) > M } . For all δ >
0, we have P ( n (ˆ τ n − τ α ∗ ) > M ) ≤ P (cid:32) sup τ ∈ D αn,M |M αn ( τ : ˆ α , ˆ α ) | [ nτ ] − [ nτ α ∗ ] ≥ δ (cid:33) + P (cid:18) inf τ ∈ D αn,M A αn ( τ : ˆ α , ˆ α )[ nτ ] − [ nτ α ∗ ] ≤ δ (cid:19) + P (cid:32) sup τ ∈ D αn,M | (cid:37) αn ( τ : ˆ α , ˆ α ) | [ nτ ] − [ nτ α ∗ ] ≥ δ (cid:33) =: P α ,n + P α ,n + P α ,n . [i] Evaluation of P α ,n . For large n , we have P α ,n = P (cid:32) sup τ ∈ D αn,M |M αn ( τ : ˆ α , ˆ α ) | [ nτ ] − [ nτ α ∗ ] ≥ δ (cid:33) ≤ P (cid:32) sup τ ∈ D αn,M |M αn ( τ : ˆ α , ˆ α ) | [ nτ ] − [ nτ α ∗ ] ≥ δ, ˆ α ∈ O α ∗ , ˆ α ∈ O α ∗ (cid:33) + P (ˆ α / ∈ O α ∗ ) + P (ˆ α / ∈ O α ∗ ) ≤ P (cid:32) sup τ ∈ D αn,M sup α k ∈O α ∗ k |M αn ( τ : α , α ) | [ nτ ] − [ nτ α ∗ ] ≥ δ (cid:33) + P (ˆ α / ∈ O α ∗ ) + P (ˆ α / ∈ O α ∗ ) . By the uniform version on the H´ajek-Renyi inequality in Lemma 2 of Iacus and Yoshida (2012), we obtain P (cid:32) sup τ ∈ D αn,M nτ ] − [ nτ α ∗ ] sup α k ∈O α ∗ k |M αn ( τ : α , α ) | ≥ δ (cid:33) ≤ P max j>M − j sup α k ∈O α ∗ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ α ∗ ]+ j (cid:88) i =[ nτ α ∗ ]+1 (cid:0) F i ( α ) − F i ( α ) − E [ F i ( α ) − F i ( α ) | G ni − ] (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ δ ≤ (cid:88) j>M − C ( δj ) ≤ C (cid:48) δ M =: γ α ( M ) . From [C6-I] , we have P (ˆ α k / ∈ O α ∗ k ) < (cid:15) for large n . Therefore P α ,n ≤ γ α ( M ) + 2 (cid:15) for large n . STIMATION FOR CHANGE POINT OF ERGODIC DIFFUSION PROCESSES 33 [ii] Evaluation of P α ,n . If ˆ α k ∈ O α ∗ k , then there exists a positive constant c independent of i such thatΓ αi − (ˆ α , ˆ α ) = Γ αi − ( α ∗ , α ∗ ) + (cid:90) ∂ ( α ,α ) Γ αi − ( α , α ) (cid:12)(cid:12) α k = α ∗ k + u (ˆ α k − α ∗ k ) d u (cid:18) ˆ α − α ∗ ˆ α − α ∗ (cid:19) ≥ Γ αi − ( α ∗ , α ∗ ) − c ( | ˆ α − α ∗ | + | ˆ α − α ∗ | ) . According to [B1-I] , if we set δ = 14 inf x Γ α ( x, α ∗ , α ∗ ) > , then for large n , P α ,n ≤ P (cid:18) inf τ ∈ D αn,M A αn ( τ : ˆ α , ˆ α )[ nτ ] − [ nτ α ∗ ] ≤ δ, ˆ α ∈ O α ∗ , ˆ α ∈ O α ∗ (cid:19) + P (cid:0) ˆ α / ∈ O α ∗ (cid:1) + P (cid:0) ˆ α / ∈ O α ∗ (cid:1) ≤ P inf τ ∈ D αn,M nτ ] − [ nτ α ∗ ] [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 (cid:16) Γ αi − ( α ∗ , α ∗ ) − c ( | ˆ α − α ∗ | + | ˆ α − α ∗ | ) (cid:17) ≤ δ + P (cid:0) ˆ α / ∈ O α ∗ (cid:1) + P (cid:0) ˆ α / ∈ O α ∗ (cid:1) ≤ P inf τ ∈ D αn,M nτ ] − [ nτ α ∗ ] [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 Γ αi − ( α ∗ , α ∗ ) − c ( | ˆ α − α ∗ | + | ˆ α − α ∗ | ) ≤ δ + P (cid:0) ˆ α / ∈ O α ∗ (cid:1) + P (cid:0) ˆ α / ∈ O α ∗ (cid:1) ≤ P inf τ ∈ D αn,M nτ ] − [ nτ α ∗ ] [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 Γ αi − ( α ∗ , α ∗ ) ≤ δ + P (cid:16) − c ( | ˆ α − α ∗ | + | ˆ α − α ∗ | ) ≤ − δ (cid:17) + P (cid:0) ˆ α / ∈ O α ∗ (cid:1) + P (cid:0) ˆ α / ∈ O α ∗ (cid:1) ≤ P (cid:16) inf x Γ α ( x, α ∗ , α ∗ ) ≤ δ (cid:17) + P (cid:18) | ˆ α − α ∗ | + | ˆ α − α ∗ | ≥ δc (cid:19) + P (cid:0) ˆ α / ∈ O α ∗ (cid:1) + P (cid:0) ˆ α / ∈ O α ∗ (cid:1) ≤ (cid:15) thanks to P (cid:18) | ˆ α − α ∗ | + | ˆ α − α ∗ | ≥ δc (cid:19) ≤ P (cid:18) | ˆ α − α ∗ | ≥ δ c (cid:19) + P (cid:18) | ˆ α − α ∗ | ≥ δ c (cid:19) ≤ (cid:15) from [C6-I] .[iii] Evaluation of P α ,n .tr (cid:16)(cid:0) A − i − (ˆ α ) − A − i − (ˆ α ) (cid:1) (cid:0) A i − (ˆ α ) − h − E α ∗ [(∆ X i ) ⊗ | G ni − ] (cid:1)(cid:17) = tr (cid:16)(cid:0) A − i − (ˆ α ) − A − i − (ˆ α ) (cid:1) (cid:0) A − i − (ˆ α ) − A − i − ( α ∗ ) − hQ i − ( θ ∗ ) + R i − ( h , θ ) (cid:1)(cid:17) ≤ (cid:90) (cid:104) tr (cid:16) ( A − i − (ˆ α ) − A − i − (ˆ α )) ∂ α (cid:96) A i − ( α ) (cid:17)(cid:105) (cid:96) (cid:12)(cid:12)(cid:12) α = α ∗ + u (ˆ α − α ∗ ) d u (ˆ α − α ∗ )+ h | A − i − (ˆ α ) − A − i − (ˆ α ) || Q i − ( θ ∗ ) | + R i − ( h , θ )and sup τ ∈ D αn,M | (cid:37) αn ( τ : ˆ α , ˆ α ) | [ nτ ] − [ nτ α ∗ ] ≤ √ n sup τ ∈ D αn,M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) nτ ] − [ nτ α ∗ ] [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 (cid:90) (cid:104) tr (cid:16) ( A − i − (ˆ α ) − A − i − (ˆ α )) ∂ α (cid:96) A i − ( α ) (cid:17)(cid:105) (cid:96) (cid:12)(cid:12)(cid:12) α = α ∗ + u (ˆ α − α ∗ ) d u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × √ n | ˆ α − α ∗ | + sup τ ∈ D αn,M h [ nτ ] − [ nτ α ∗ ] [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 | A − i − (ˆ α ) − A − i − (ˆ α ) || Q i − ( θ ∗ ) | + sup τ ∈ D αn,M h [ nτ ] − [ nτ α ∗ ] [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 R i − (1 , θ ) ≤ √ n sup x,α k (cid:12)(cid:12)(cid:12)(cid:104) tr (cid:16) ( A − ( x, α ) − A − ( x, α )) ∂ α (cid:96) A ( x, α ) (cid:17)(cid:105) (cid:96) (cid:12)(cid:12)(cid:12) √ n | ˆ α − α ∗ | + h sup x,α k (cid:12)(cid:12) A − ( x, α ) − A − ( x, α ) (cid:12)(cid:12) sup x,θ | Q ( x, θ ) | + h M n (cid:88) i =[ nτ α ∗ ]+1 R i − (1 , θ )= o p (1) . Hence, we see P α ,n ≤ P (cid:32) sup τ ∈ D αn,M | (cid:37) αn ( τ : ˆ α , ˆ α ) | [ nτ ] − [ nτ α ∗ ] ≥ δ, ˆ α ∈ O α ∗ , ˆ α ∈ O α ∗ (cid:33) + P (ˆ α / ∈ O α ∗ ) + P (ˆ α / ∈ O α ∗ ) ≤ (cid:15) for large n .[iv] From the evaluations in Steps [i]-[iii], we havelim n →∞ P ( n (ˆ τ αn − τ α ∗ ) > M ) ≤ γ α ( M ) + 9 (cid:15) for any M ≥ (cid:15) >
0. Therefore lim M →∞ lim n →∞ P ( n (ˆ τ αn − τ α ∗ ) > M ) ≤ (cid:15). (cid:3) Proof of Corollary 1.
It is sufficient to show, for all (cid:15) ∈ [0 ,
1) and
M > n →∞ P ( n (cid:15) (ˆ τ αn − τ α ∗ ) > M ) = 0 . Let D αn,M = { τ ∈ [0 , | n (cid:15) ( τ − τ α ∗ ) > M } . Suppose that there exists δ ∈ (0 , − (cid:15) ) such that nh / ( (cid:15) + δ ) −→
0. We have P ( n (cid:15) (ˆ τ n − τ α ∗ ) > M ) ≤ P (cid:32) sup τ ∈ D αn,M |M αn ( τ : ˆ α , ˆ α ) | n − δ ([ nτ ] − [ nτ α ∗ ]) ≥ (cid:33) + P (cid:18) inf τ ∈ D αn,M A αn ( τ : ˆ α , ˆ α ) n − δ ([ nτ ] − [ nτ α ∗ ]) ≤ (cid:19) + P (cid:32) sup τ ∈ D αn,M | (cid:37) αn ( τ : ˆ α , ˆ α ) | n − δ ([ nτ ] − [ nτ α ∗ ]) ≥ (cid:33) =: P α ,n + P α ,n + P α ,n . [i] Evaluation of P α ,n . For large n , we have P α ,n = P (cid:32) sup τ ∈ D αn,M |M αn ( τ : ˆ α , ˆ α ) | n − δ ([ nτ ] − [ nτ α ∗ ]) ≥ (cid:33) ≤ P (cid:32) sup τ ∈ D αn,M |M αn ( τ : ˆ α , ˆ α ) | n − δ ([ nτ ] − [ nτ α ∗ ]) ≥ , ˆ α ∈ O α ∗ , ˆ α ∈ O α ∗ (cid:33) + P (ˆ α / ∈ O α ∗ ) + P (ˆ α / ∈ O α ∗ ) ≤ P (cid:32) sup τ ∈ D αn,M sup α k ∈O α ∗ k |M αn ( τ : α , α ) | [ nτ ] − [ nτ α ∗ ] ≥ n − δ (cid:33) + P (ˆ α / ∈ O α ∗ ) + P (ˆ α / ∈ O α ∗ ) . By the uniform version on the H´ajek-Renyi inequality in Lemma 2 of Iacus and Yoshida (2012), we obtain P (cid:32) sup τ ∈ D αn,M nτ ] − [ nτ α ∗ ] sup α k ∈O α ∗ k |M αn ( τ : α , α ) | ≥ n − δ (cid:33) STIMATION FOR CHANGE POINT OF ERGODIC DIFFUSION PROCESSES 35 ≤ P max j>M/n (cid:15) − − j sup α k ∈O α ∗ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ α ∗ ]+ j (cid:88) i =[ nτ α ∗ ]+1 (cid:0) F i ( α ) − F i ( α ) − E [ F i ( α ) − F i ( α ) | G ni − ] (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ n − δ ≤ (cid:88) j>M/n (cid:15) − − C ( n − δ j ) ≤ C (cid:48) n − δ n (cid:15) − M = C (cid:48) M n (cid:15) +2 δ − −→ . From [C6-I] , we have P (ˆ α k / ∈ O α ∗ k ) < (cid:15) for large n . Therefore P α ,n ≤ (cid:15) for large n .[ii] Evaluation of P α ,n . If ˆ α k ∈ O α ∗ k , then there exists a positive constant c independent of i such thatΓ αi − (ˆ α , ˆ α ) = Γ αi − ( α ∗ , α ∗ ) + (cid:90) ∂ ( α ,α ) Γ αi − ( α , α ) (cid:12)(cid:12) α k = α ∗ k + u (ˆ α k − α ∗ k ) d u (cid:18) ˆ α − α ∗ ˆ α − α ∗ (cid:19) ≥ Γ αi − ( α ∗ , α ∗ ) − c ( | ˆ α − α ∗ | + | ˆ α − α ∗ | ) . We see, for large n , P α ,n ≤ P (cid:18) inf τ ∈ D αn,M A αn ( τ : ˆ α , ˆ α ) n − δ ([ nτ ] − [ nτ α ∗ ]) ≤ , ˆ α ∈ O α ∗ , ˆ α ∈ O α ∗ (cid:19) + P (cid:0) ˆ α / ∈ O α ∗ (cid:1) + P (cid:0) ˆ α / ∈ O α ∗ (cid:1) ≤ P inf τ ∈ D αn,M nτ ] − [ nτ α ∗ ] [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 (cid:16) Γ αi − ( α ∗ , α ∗ ) − c ( | ˆ α − α ∗ | + | ˆ α − α ∗ | ) (cid:17) ≤ n − δ + P (cid:0) ˆ α / ∈ O α ∗ (cid:1) + P (cid:0) ˆ α / ∈ O α ∗ (cid:1) ≤ P inf τ ∈ D αn,M nτ ] − [ nτ α ∗ ] [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 Γ αi − ( α ∗ , α ∗ ) − c ( | ˆ α − α ∗ | + | ˆ α − α ∗ | ) ≤ n − δ + P (cid:0) ˆ α / ∈ O α ∗ (cid:1) + P (cid:0) ˆ α / ∈ O α ∗ (cid:1) ≤ P inf τ ∈ D αn,M nτ ] − [ nτ α ∗ ] [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 Γ αi − ( α ∗ , α ∗ ) ≤ n − δ + P (cid:16) − c ( | ˆ α − α ∗ | + | ˆ α − α ∗ | ) ≤ − n − δ (cid:17) + P (cid:0) ˆ α / ∈ O α ∗ (cid:1) + P (cid:0) ˆ α / ∈ O α ∗ (cid:1) ≤ P (cid:16) inf x Γ α ( x, α ∗ , α ∗ ) ≤ n − δ (cid:17) + P (cid:18) | ˆ α − α ∗ | + | ˆ α − α ∗ | ≥ n − δ c (cid:19) + P (cid:0) ˆ α / ∈ O α ∗ (cid:1) + P (cid:0) ˆ α / ∈ O α ∗ (cid:1) ≤ (cid:15) thanks to P (cid:18) | ˆ α − α ∗ | + | ˆ α − α ∗ | ≥ n − δ c (cid:19) ≤ P (cid:18) n δ − / √ n | ˆ α − α ∗ | ≥ c (cid:19) + P (cid:18) n δ − / √ n | ˆ α − α ∗ | ≥ c (cid:19) ≤ (cid:15) from [C6-I] .[iii] Evaluation of P α ,n .tr (cid:16)(cid:0) A − i − (ˆ α ) − A − i − (ˆ α ) (cid:1) (cid:0) A i − (ˆ α ) − h − E α ∗ [(∆ X i ) ⊗ | G ni − ] (cid:1)(cid:17) = tr (cid:16)(cid:0) A − i − (ˆ α ) − A − i − (ˆ α ) (cid:1) ( A i − (ˆ α ) − A i − ( α ∗ )) (cid:17) + R i − ( h, θ ) ≤ (cid:90) (cid:104) tr (cid:16) ( A − i − (ˆ α ) − A − i − (ˆ α )) ∂ α (cid:96) A i − ( α ) (cid:17)(cid:105) (cid:96) (cid:12)(cid:12)(cid:12) α = α ∗ + u (ˆ α − α ∗ ) d u (ˆ α − α ∗ ) + R i − ( h, θ )andsup τ ∈ D αn,M | (cid:37) αn ( τ : ˆ α , ˆ α ) | n − δ ([ nτ ] − [ nτ α ∗ ]) ≤ n − δ +1 / sup τ ∈ D αn,M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) nτ ] − [ nτ α ∗ ] [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 (cid:90) (cid:104) tr (cid:16) ( A − i − (ˆ α ) − A − i − (ˆ α )) ∂ α (cid:96) A i − ( α ) (cid:17)(cid:105) (cid:96) (cid:12)(cid:12)(cid:12) α = α ∗ + u (ˆ α − α ∗ ) d u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × √ n | ˆ α − α ∗ | + hn − δ sup τ ∈ D αn,M nτ ] − [ nτ α ∗ ] [ nτ ] (cid:88) i =[ nτ α ∗ ]+1 R i − (1 , θ ) ≤ n − δ +1 / sup x,α k (cid:12)(cid:12)(cid:12)(cid:104) tr (cid:16) ( A − ( x, α ) − A − ( x, α )) ∂ α (cid:96) A ( x, α ) (cid:17)(cid:105) (cid:96) (cid:12)(cid:12)(cid:12) √ n | ˆ α − α ∗ | + hn − δ n (cid:15) − M n (cid:88) i =[ nτ α ∗ ]+1 R i − (1 , θ )= O p ( n δ − / ) + O p ( n (cid:15) + δ h ) = O p ( n δ − / ) + O p ( nh / ( (cid:15) + δ ) ) = o p (1) . Hence, we see P α ,n ≤ P (cid:32) sup τ ∈ D αn,M | (cid:37) αn ( τ : ˆ α , ˆ α ) | n − δ ([ nτ ] − [ nτ α ∗ ]) ≥ , ˆ α ∈ O α ∗ , ˆ α ∈ O α ∗ (cid:33) + P (ˆ α / ∈ O α ∗ ) + P (ˆ α / ∈ O α ∗ ) ≤ (cid:15) for large n .[iv] From the evaluations in Steps [i]-[iii], we havelim n →∞ P ( n (cid:15) (ˆ τ αn − τ α ∗ ) > M ) ≤ (cid:15) for any M > (cid:15) > (cid:3) In Case A of Situation II, we set G n ( v ) = Ψ n (cid:32) τ β ∗ + vT ϑ β : β ∗ , β ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α ∗ (cid:33) − Ψ n (cid:0) τ β ∗ : β ∗ , β ∗ (cid:12)(cid:12) α ∗ (cid:1) , ˆ G n ( v ) = Ψ n (cid:32) τ β ∗ + vT ϑ β : ˆ β , ˆ β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ α (cid:33) − Ψ n (cid:16) τ β ∗ : ˆ β , ˆ β (cid:12)(cid:12)(cid:12) ˆ α (cid:17) , D βn ( v ) = ˆ G n ( v ) − G n ( v ) . Lemma 4
Suppose that [C1] - [C5] , [C6-II] , [A1-II] - [A3-II] hold. Then, for all L > v ∈ [ − L,L ] |D βn ( v ) | p −→ n → ∞ . Proof.
We assume that v >
0. Then, we can express D βn ( v ) = [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:16) G i ( ˆ β | ˆ α ) − G i ( ˆ β | ˆ α ) (cid:17) − [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:16) G i ( β ∗ | α ∗ ) − G i ( β ∗ | α ∗ ) (cid:17) = [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:18) G i ( ˆ β | α ∗ ) − G i ( ˆ β | α ∗ ) + (cid:90) (cid:16) ∂ α G i ( ˆ β | α ) − ∂ α G i ( ˆ β | α ) (cid:17)(cid:12)(cid:12)(cid:12) α = α ∗ + u (ˆ α − α ∗ ) d u (ˆ α − α ∗ ) (cid:19) − [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:16) G i ( β ∗ | α ∗ ) − G i ( β ∗ | α ∗ ) (cid:17) = [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:32) G i ( β ∗ | α ∗ ) − G i ( β ∗ | α ∗ ) + m − (cid:88) j =1 j ! (cid:16) ∂ jβ G i ( β ∗ | α ∗ ) ⊗ ( ˆ β − β ∗ ) ⊗ j − ∂ jβ G i ( β ∗ | α ∗ ) ⊗ ( ˆ β − β ∗ ) ⊗ j (cid:17) STIMATION FOR CHANGE POINT OF ERGODIC DIFFUSION PROCESSES 37 + (cid:90) (1 − u ) m − ( m − ∂ mβ G i ( β ∗ + u ( ˆ β − β ∗ ) | α ∗ )d u ⊗ ( ˆ β − β ∗ ) ⊗ m − (cid:90) (1 − u ) m − ( m − ∂ mβ G i ( β ∗ + u ( ˆ β − β ∗ ) | α ∗ )d u ⊗ ( ˆ β − β ∗ ) ⊗ m + (cid:90) (cid:90) ∂ β ∂ α G i ( ˆ β + v ( ˆ β − ˆ β ) | α ∗ + u (ˆ α − α ∗ ))d u d v ⊗ (ˆ α − α ∗ ) ⊗ ( ˆ β − ˆ β ) (cid:33) − [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:16) G i ( β ∗ | α ∗ ) − G i ( β ∗ | α ∗ ) (cid:17) = [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 m − (cid:88) j =1 j ! (cid:16) ∂ jβ G i ( β ∗ | α ∗ ) ⊗ ( ˆ β − β ∗ ) ⊗ j − ∂ jβ G i ( β ∗ | α ∗ ) ⊗ ( ˆ β − β ∗ ) ⊗ j (cid:17) + [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:18)(cid:90) (1 − u ) m − ( m − ∂ mβ G i ( β ∗ + u ( ˆ β − β ∗ ) | α ∗ )d u ⊗ ( ˆ β − β ∗ ) ⊗ m − (cid:90) (1 − u ) m − ( m − ∂ mβ G i ( β ∗ + u ( ˆ β − β ∗ ) | α ∗ )d u ⊗ ( ˆ β − β ∗ ) ⊗ m + (cid:90) (cid:90) ∂ β ∂ α G i ( ˆ β + v ( ˆ β − ˆ β ) | α ∗ + u (ˆ α − α ∗ ))d u d v ⊗ (ˆ α − α ∗ ) ⊗ ( ˆ β − ˆ β ) (cid:19) . (7.29)Now we see sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:90) (1 − u ) m − ( m − ∂ mβ G i ( β ∗ j + u ( ˆ β j − β ∗ j ) | α ∗ )d u ⊗ ( ˆ β k − β ∗ k ) ⊗ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ [ nτ β ∗ + L/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) (1 − u ) m − ( m − ∂ mβ G i ( β ∗ k + u ( ˆ β k − β ∗ k ) | α ∗ )d u (cid:12)(cid:12)(cid:12)(cid:12) | ˆ β k − β ∗ k | m ≤ T m/ nτ β ∗ + L/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 sup β ∈ Θ B (cid:12)(cid:12) ∂ mβ G i ( β | α ∗ ) (cid:12)(cid:12) (cid:16) √ T | ˆ β k − β ∗ k | (cid:17) m = O p (cid:32) T m/ h / ϑ β (cid:33) = O p (cid:32) n m/ − h ( m − / T ϑ β (cid:33) = o p (1) , (7.30)sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:90) (cid:90) ∂ β ∂ α G i ( ˆ β + v ( ˆ β − ˆ β ) | α ∗ + u (ˆ α − α ∗ ))d u d v ⊗ (ˆ α − α ∗ ) ⊗ ( ˆ β − ˆ β ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ [ nτ β ∗ + L/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) (cid:90) ∂ β ∂ α G i ( ˆ β + v ( ˆ β − ˆ β ) | α ∗ + u (ˆ α − α ∗ ))d u d v (cid:12)(cid:12)(cid:12)(cid:12) | ˆ α − α ∗ || ˆ β − ˆ β |≤ ϑ β √ n [ nτ β ∗ + L/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 sup ( α,β ) ∈ Θ | ∂ β ∂ α G i ( β | α ) | ϑ − β | ˆ β − ˆ β |√ n | ˆ α − α ∗ | = O p (cid:32) √ T ϑ β (cid:33) = o p (1) (7.31) and [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 ∂ jβ G i ( β ∗ k | α ∗ ) ⊗ ( ˆ β k − β ∗ k ) ⊗ j = [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:16) ∂ jβ G i ( β ∗ k | α ∗ ) − E β ∗ [ ∂ jβ G i ( β ∗ k | α ∗ ) | G ni − ] (cid:17) ⊗ ( ˆ β k − β ∗ k ) ⊗ j + [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 E β ∗ [ ∂ jβ G i ( β ∗ k | α ∗ ) | G ni − ] ⊗ ( ˆ β k − β ∗ k ) ⊗ j . (7.32)By Theorem 2.11 of Hall and Heyde (1980), we have E β ∗ T j sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:16) ∂ jβ G i ( β ∗ k | α ∗ ) − E β ∗ [ ∂ jβ G i ( β ∗ k | α ∗ ) | G ni − ] (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ CT j [ nτ β ∗ + L/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 E β ∗ (cid:20) E β ∗ (cid:20)(cid:12)(cid:12)(cid:12) ∂ jβ G i ( β ∗ k | α ∗ ) − E [ ∂ jβ G i ( β ∗ k | α ∗ ) | G ni − ] (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) G ni − (cid:21)(cid:21) ≤ C (cid:48) T hhϑ β = C (cid:48) T ϑ β −→ T j/ sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:16) ∂ jβ G i ( β ∗ k | α ∗ ) − E β ∗ [ ∂ jβ G i ( β ∗ k | α ∗ ) | G ni − ] (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = o p (1) . (7.33)Moreover, we seesup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 E β ∗ [ ∂ β G i ( β ∗ k | α ∗ ) | G ni − ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | ˆ β k − β ∗ k | = sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:18) − h (cid:104) ∂ β (cid:96) b i − ( β ∗ k ) T A − i − ( α ∗ )( b i − ( β ∗ ) − b i − ( β ∗ k )) (cid:105) (cid:96) + R i − ( h , θ ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | ˆ β k − β ∗ k | = sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:18) − h (cid:16) Ξ βi − ( α ∗ , β ∗ k )( β ∗ − β ∗ k )+ (cid:90) (1 − u ) (cid:104) ∂ β (cid:96) b i − ( β ∗ k ) T A − i − ( α ∗ ) ∂ β (cid:96) ∂ β (cid:96) b i − ( β ∗ k + u ( β ∗ − β ∗ k )) (cid:105) (cid:96) ,(cid:96) ,(cid:96) d u ⊗ ( β ∗ − β ∗ k ) ⊗ (cid:17) + R i − ( h , θ ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | ˆ β k − β ∗ k | = sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:18) − h (cid:16) Ξ βi − ( α ∗ , β )( β ∗ − β ∗ k )+ (cid:90) ∂ β Ξ βi − ( α ∗ , β + u ( β ∗ k − β ))d u ⊗ ( β ∗ − β ∗ k ) ⊗ ( β ∗ k − β )+ (cid:90) (1 − u ) (cid:104) ∂ β (cid:96) b i − ( β ∗ k ) T A − i − ( α ∗ ) ∂ β (cid:96) ∂ β (cid:96) b i − ( β ∗ k + u ( β ∗ − β ∗ k )) (cid:105) (cid:96) ,(cid:96) ,(cid:96) d u ⊗ ( β ∗ − β ∗ k ) ⊗ (cid:17) STIMATION FOR CHANGE POINT OF ERGODIC DIFFUSION PROCESSES 39 + R i − ( h , θ ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | ˆ β k − β ∗ k |≤ hϑ β √ T sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 Ξ βi − ( α ∗ , β ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϑ − β | β ∗ − β ∗ k |√ T | ˆ β k − β ∗ k | + hϑ β √ T [ nτ β ∗ + L/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 R i − (1 , θ ) ϑ − β | β ∗ − β ∗ k | ϑ − β | β ∗ k − β |√ T | ˆ β k − β ∗ k | + hϑ β √ T [ nτ β ∗ + L/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 R i − (1 , θ )( ϑ − β | β ∗ − β ∗ k | ) √ T | ˆ β k − β ∗ k | + 1 √ T [ nτ β ∗ + L/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 R i − ( h , θ ) √ T | ˆ β k − β ∗ k | = O p (cid:32) √ T ϑ β (cid:33) + O p (cid:18) √ T (cid:19) + O p (cid:32) h √ T ϑ β (cid:33) = o p (1) (7.34)and, for j ≥ v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 E β ∗ [ ∂ jβ G i ( β ∗ k | α ∗ ) | G ni − ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | ˆ β k − β ∗ k | j ≤ [ nτ β ∗ + L/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 R i − ( h, θ ) | ˆ β j − β ∗ j | j = hT j/ nτ β ∗ + L/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 R i − (1 , θ ) (cid:16) √ T | ˆ β k − β ∗ k | (cid:17) j = O p (cid:32) T ϑ β (cid:33) = o p (1) . (7.35)Therefore, from (7.29)-(7.35), we have sup v ∈ [0 ,L ] |D βn ( v ) | p −→ . By the similar proof, we see sup v ∈ [ − L, |D βn ( v ) | p −→
0, and this proof is complete. (cid:3)
Lemma 5
Suppose that [C1] - [C5] , [C6-II] , [A1-II] - [A3-II] hold. Then, for all L > G n ( v ) w −→ G ( v ) in D [ − L, L ]as n → ∞ . Proof.
We consider v >
0. We have G n ( v ) = [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:16) G i ( β ∗ | α ∗ ) − G i ( β ∗ | α ∗ ) (cid:17) = [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:32) m − (cid:88) j =1 j ! ∂ jβ G i ( β ∗ | α ∗ ) ⊗ ( β ∗ − β ∗ ) ⊗ j + (cid:90) (1 − u ) m − ( m − ∂ mβ G i ( β ∗ + u ( β ∗ − β ∗ ) | α ∗ )d u ⊗ ( β ∗ − β ∗ ) ⊗ m (cid:33) = [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:18) ∂ β G i ( β ∗ | α ∗ )( β ∗ − β ∗ ) + 12 ∂ β G i ( β ∗ | α ∗ ) ⊗ ( β ∗ − β ∗ ) ⊗ (cid:19) + [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:32) m − (cid:88) j =3 j ! ∂ jβ G i ( β ∗ | α ∗ ) ⊗ ( β ∗ − β ∗ ) ⊗ j + (cid:90) (1 − u ) m − ( m − ∂ mβ G i ( β ∗ + u ( β ∗ − β ∗ ) | α ∗ )d u ⊗ ( β ∗ − β ∗ ) ⊗ m (cid:33) = [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:18) ∂ β G i ( β ∗ | α ∗ )( β ∗ − β ∗ ) + 12 ∂ β G i ( β ∗ | α ∗ ) ⊗ ( β ∗ − β ∗ ) ⊗ (cid:19) + ¯ o p (1)=: G ,n ( v ) + G ,n ( v ) + ¯ o p (1) , where, for j ≥ ϑ jβ sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:16) ∂ jβ G i ( β ∗ | α ∗ ) − E [ ∂ jβ G i ( β ∗ | α ∗ ) | G ni − ] (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = o p (1) ,ϑ jβ sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 E β ∗ [ ∂ jβ G i ( β ∗ | α ∗ ) | G ni − ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = o p (1) , sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:90) (1 − u ) m − ( m − ∂ mβ G i ( β ∗ + u ( β ∗ − β ∗ ) | α ∗ )d u ⊗ ( β ∗ − β ∗ ) ⊗ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = o p (1) . Now, we see G ,n ( v ) = [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:16) ∂ β G i ( β ∗ | α ∗ ) − E β ∗ [ ∂ β G i ( β ∗ | α ∗ ) | G ni − ] (cid:17) ( β ∗ − β ∗ )+ [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 E β ∗ [ ∂ β G i ( β ∗ | α ∗ ) | G ni − ]( β ∗ − β ∗ ) , (7.36)sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 E β ∗ [ ∂ β G i ( β ∗ | α ∗ ) | G ni − ]( β ∗ − β ∗ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ϑ β [ nτ β ∗ + L/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 R i − ( h , θ ) = O p (cid:18) hϑ β (cid:19) = o p (1) , (7.37) [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 E β ∗ (cid:20)(cid:16)(cid:0) ∂ β G i ( β ∗ | α ∗ ) − E β ∗ [ ∂ β G i ( β ∗ | α ∗ ) | G ni − ] (cid:1) ( β ∗ − β ∗ ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) G ni − (cid:21) = ( β ∗ − β ∗ ) T [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 E β ∗ (cid:20)(cid:16) ∂ β G i ( β ∗ | α ∗ ) − E β ∗ [ ∂ β G i ( β ∗ | α ∗ ) | G ni − ] (cid:17) T (cid:16) ∂ β G i ( β ∗ | α ∗ ) − E β ∗ [ ∂ β G i ( β ∗ | α ∗ ) | G ni − ] (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) G ni − (cid:21) ( β ∗ − β ∗ )= ( β ∗ − β ∗ ) T [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:16) h Ξ βi − ( α ∗ , β ∗ ) + R i − ( h , θ ) (cid:17) ( β ∗ − β ∗ ) p −→ e T β (cid:90) R d Ξ β ( x, α ∗ , β )d µ ( α ∗ ,β ) ( x ) e β v = 4 J β v (7.38) STIMATION FOR CHANGE POINT OF ERGODIC DIFFUSION PROCESSES 41 and [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 E (cid:20)(cid:16)(cid:0) ∂ β G i ( β ∗ | α ∗ ) − E β ∗ [ ∂ β G i ( β ∗ | α ∗ ) | G ni − ] (cid:1) ( β ∗ − β ∗ ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) G ni − (cid:21) = [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 ϑ β R i − ( h , θ ) p −→ . (7.39)According to Corollary 3.8 of McLeish (1974), we obtain, from (7.38) and (7.39), [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:16) ∂ β G i ( β ∗ | α ∗ ) − E β ∗ [ ∂ β G i ( β ∗ | α ∗ ) | G ni − ] (cid:17) ( β ∗ − β ∗ ) w −→ − J / β W ( v ) in D [0 , L ] . (7.40)Further, from (7.36), (7.37) and (7.40), we have G ,n ( v ) w −→ − J / β W ( v ) in D [0 , L ].Besides, we seesup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 ∂ β G i ( β ∗ | α ∗ ) ⊗ ( β ∗ − β ∗ ) ⊗ − J β v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:16) ∂ β G i ( β ∗ | α ∗ ) − E β ∗ (cid:2) ∂ β G i ( β ∗ | α ∗ ) (cid:12)(cid:12) G ni − (cid:3)(cid:17) ⊗ ( β ∗ − β ∗ ) ⊗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 E β ∗ (cid:2) ∂ β G i ( β ∗ | α ∗ ) (cid:12)(cid:12) G ni − (cid:3) ⊗ ( β ∗ − β ∗ ) ⊗ − J β v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (7.41)sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:16) ∂ β G i ( β ∗ | α ∗ ) − E β ∗ (cid:2) ∂ β G i ( β ∗ | α ∗ ) (cid:12)(cid:12) G ni − (cid:3)(cid:17) ⊗ ( β ∗ − β ∗ ) ⊗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p −→ v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 E β ∗ (cid:2) ∂ β G i ( β ∗ | α ∗ ) (cid:12)(cid:12) G ni − (cid:3) ⊗ ( β ∗ − β ∗ ) ⊗ − J β v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:16) h Ξ βi − ( α ∗ , β ∗ ) + R i − ( h , θ ) (cid:17) ⊗ ( β ∗ − β ∗ ) ⊗ − J β v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 h Ξ βi − ( α ∗ , β ) ⊗ ( β ∗ − β ∗ ) ⊗ − J β v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + o p (1) p −→ , (7.43)where (7.42) is obtained by E β ∗ sup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 (cid:0) ∂ β G i ( β ∗ | α ∗ ) − E β ∗ (cid:2) ∂ β G i ( β ∗ | α ∗ ) (cid:12)(cid:12) G ni − (cid:3)(cid:1) ⊗ ( β ∗ − β ∗ ) ⊗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Cϑ β [ nτ β ∗ + L/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 E β ∗ (cid:104)(cid:12)(cid:12) ∂ β G i ( β ∗ | α ∗ ) − E β ∗ (cid:2) ∂ β G i ( β ∗ | α ∗ ) (cid:12)(cid:12) G ni − (cid:3)(cid:12)(cid:12) (cid:105) ≤ C (cid:48) ϑ β −→ from Theorem 2.11 of Hall and Heyde (1980), and (7.43) is obtained bysup v ∈ [0 ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 h Ξ βi − ( α ∗ , β ) ⊗ ( β ∗ − β ∗ ) ⊗ − J β v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ sup v ∈ [0 ,(cid:15) n ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 h Ξ βi − ( α ∗ , β ) ⊗ ( β ∗ − β ∗ ) ⊗ − J β v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + sup v ∈ [ (cid:15) n ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 h Ξ βi − ( α ∗ , β ) ⊗ ( β ∗ − β ∗ ) ⊗ − J β v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:15) n sup v ∈ [0 ,(cid:15) n ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 h Ξ βi − ( α ∗ , β ) ⊗ ( β ∗ − β ∗ ) ⊗ − J β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + L sup v ∈ [ (cid:15) n ,L ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v [ nτ β ∗ + v/hϑ β ] (cid:88) i =[ nτ β ∗ ]+1 h Ξ βi − ( α ∗ , β ) ⊗ ( β ∗ − β ∗ ) ⊗ − J β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ O p ( (cid:15) n ) + L max [ n /r ] ≤ k ≤ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k [ nτ β ∗ ]+ k (cid:88) i =[ nτ β ∗ ]+1 h Ξ βi − ( α ∗ , β ) ⊗ ( ϑ − β ( β ∗ − β ∗ )) ⊗ − J β (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = o p (1)Here { (cid:15) n } ∞ n =1 is a positive sequence such that (cid:15) n −→ (cid:15) n /ϑ β −→ ∞ , and r is a constant with r ∈ (1 , nh r −→ ∞ . From (7.41)-(7.43), we have sup v ∈ [0 ,L ] | G ,n ( v ) − J β v | p −→
0. Therefore we obtain G n ( v ) w −→ − J / β W ( v ) + J β v in D [0 , L ] . The argument for v < (cid:3)
Proof of Theorem 3.
It is enough to showlim M →∞ lim n →∞ P (cid:0) T ϑ β (ˆ τ βn − τ β ∗ ) > M (cid:1) = 0 . (7.44)Because in the same way as the proof of Theorem 1, (7.44) leads to T ϑ β (ˆ τ βn − τ β ∗ ) = O p (1), and this andLemmas 1, 4, 5 yield T ϑ β (ˆ τ βn − τ β ∗ ) d −→ arg min v ∈ R G ( v ) . We show (7.44) below. For τ > τ β ∗ , we haveΨ n ( τ : β , β | α ) − Ψ n ( τ β ∗ : β , β | α ) = [ nτ ] (cid:88) i =1 G i ( β | α ) + n (cid:88) i =[ nτ ]+1 G i ( β | α ) − [ nτ β ∗ ] (cid:88) i =1 G i ( β | α ) − n (cid:88) i =[ nτ β ∗ ]+1 G i ( β | α )= [ nτ ] (cid:88) i =[ nτ β ∗ ]+1 (cid:16) G i ( β | α ) − G i ( β | α ) (cid:17) . Now, from G i ( β | α ) − G i ( β | α ) = G i ( β | α ) − G i ( β | α ) − E β ∗ [ G i ( β | α ) − G i ( β | α ) | G ni − ]+ h tr (cid:16) A − i − ( α ) ( b i − ( β ) − b i − ( β )) ⊗ (cid:17) + 2tr (cid:16) A − i − ( α ) (cid:0) hb i − ( β ) − E β ∗ [∆ X i | G ni − ] (cid:1) ( b i − ( β ) − b i − ( β )) T (cid:17) , we see Ψ n ( τ : β , β | α ) − Ψ n ( τ β ∗ : β , β | α ) STIMATION FOR CHANGE POINT OF ERGODIC DIFFUSION PROCESSES 43 = [ nτ ] (cid:88) i =[ nτ β ∗ ]+1 (cid:16) G i ( β | α ) − G i ( β | α ) − E β ∗ [ G i ( β | α ) − G i ( β | α ) | G ni − ] (cid:17) + h [ nτ ] (cid:88) i =[ nτ β ∗ ]+1 tr (cid:16) A − i − ( α ) ( b i − ( β ) − b i − ( β )) ⊗ (cid:17) + 2 [ nτ ] (cid:88) i =[ nτ β ∗ ]+1 tr (cid:16) A − i − ( α ) (cid:0) hb i − ( β ) − E β ∗ [∆ X i | G ni − ] (cid:1) ( b i − ( β ) − b i − ( β )) T (cid:17) =: M βn ( τ : β , β | α ) + A βn ( τ : β , β | α ) + (cid:37) βn ( τ : β , β | α ) . Let M ≥ D βn,M = { τ ∈ [0 , | T ϑ β ( τ − τ β ∗ ) > M } . For all δ >
0, we have P (cid:0) T ϑ β (ˆ τ βn − τ β ∗ ) > M (cid:1) ≤ P (cid:32) inf τ ∈ D βn,M Ψ n ( τ : ˆ β , ˆ β | ˆ α ) ≤ Ψ n ( τ β ∗ : ˆ β , ˆ β | ˆ α ) (cid:33) = P (cid:32) inf τ ∈ D βn,M (cid:16) Ψ n ( τ : ˆ β , ˆ β | ˆ α ) − Ψ n ( τ β ∗ : ˆ β , ˆ β | ˆ α ) (cid:17) ≤ (cid:33) = P (cid:32) inf τ ∈ D βn,M (cid:16) M βn ( τ : ˆ β , ˆ β | ˆ α ) + A βn ( τ : ˆ β , ˆ β | ˆ α ) + (cid:37) βn ( τ : ˆ β , ˆ β | ˆ α ) (cid:17) ≤ (cid:33) ≤ P (cid:32) inf τ ∈ D βn,M M βn ( τ : ˆ β , ˆ β | ˆ α ) + A βn ( τ : ˆ β , ˆ β | ˆ α ) + (cid:37) βn ( τ : ˆ β , ˆ β | ˆ α ) hϑ β ([ nτ ] − [ nτ β ∗ ]) ≤ (cid:33) ≤ P (cid:32) inf τ ∈ D βn,M M βn ( τ : ˆ β , ˆ β | ˆ α ) hϑ β ([ nτ ] − [ nτ β ∗ ]) ≤ − δ (cid:33) + P (cid:32) inf τ ∈ D βn,M A βn ( τ : ˆ β , ˆ β | ˆ α ) hϑ β ([ nτ ] − [ nτ β ∗ ]) ≤ δ (cid:33) + P (cid:32) inf τ ∈ D βn,M (cid:37) βn ( τ : ˆ β , ˆ β | ˆ α ) hϑ β ([ nτ ] − [ nτ β ∗ ]) ≤ − δ (cid:33) ≤ P sup τ ∈ D βn,M |M βn ( τ : ˆ β , ˆ β | ˆ α ) | hϑ β ([ nτ ] − [ nτ β ∗ ]) ≥ δ + P (cid:32) inf τ ∈ D βn,M A βn ( τ : ˆ β , ˆ β | ˆ α ) hϑ β ([ nτ ] − [ nτ β ∗ ]) ≤ δ (cid:33) + P sup τ ∈ D βn,M | (cid:37) βn ( τ : ˆ β , ˆ β | ˆ α ) | hϑ β ([ nτ ] − [ nτ β ∗ ]) ≥ δ =: P β ,n + P β ,n + P β ,n . [i] Evaluation of P β ,n . Choose (cid:15) > ∂ β G i ( β | α ) is continuous with respect to β ∈ Θ B ,we can choose ¯ β ∈ O ˆ β so that M βn ( τ : ˆ β , ˆ β | α ) = [ nτ ] (cid:88) i =[ nτ β ∗ ]+1 (cid:16) G i ( ˆ β | α ) − G i ( ˆ β | α ) − E β ∗ [ G i ( ˆ β | α ) − G i ( ˆ β | α ) | G ni − ] (cid:17) = [ nτ ] (cid:88) i =[ nτ β ∗ ]+1 (cid:16) ∂ β G i ( ¯ β | α ) − E β ∗ [ ∂ β G i ( β | α ) | G ni − ] (cid:12)(cid:12) β = ¯ β (cid:17) ( ˆ β − ˆ β ) . If ˆ α ∈ O α ∗ and ˆ β k ∈ O β , then |M βn ( τ : ˆ β , ˆ β | ˆ α ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ ] (cid:88) i =[ nτ β ∗ ]+1 (cid:0) ∂ β G i ( ¯ β | ˆ α ) − E β ∗ [ ∂ β G i ( ¯ β | ˆ α ) | G ni − ] (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | ˆ β − ˆ β | ≤ sup ( α,β ) ∈ Θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ ] (cid:88) i =[ nτ β ∗ ]+1 (cid:0) ∂ β G i ( β | α ) − E β ∗ [ ∂ β G i ( β | α ) | G ni − ] (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | ˆ β − ˆ β | =: sup ( α,β ) ∈ Θ | M βn ( τ : β | α ) || ˆ β − ˆ β | . Hence we have P β ,n = P sup τ ∈ D βn,M |M βn ( τ : ˆ β , ˆ β | ˆ α ) | hϑ β ([ nτ ] − [ nτ β ∗ ]) ≥ δ ≤ P sup τ ∈ D βn,M |M βn ( τ : ˆ β , ˆ β | ˆ α ) | hϑ β ([ nτ ] − [ nτ β ∗ ]) ≥ δ, | ˆ β − ˆ β | ≤ ϑ β , ˆ α ∈ O α ∗ , ˆ β k ∈ O β + P ( | ˆ β − ˆ β | > ϑ β ) + P (ˆ α / ∈ O α ∗ ) + P ( ˆ β / ∈ O β ) + P ( ˆ β / ∈ O β ) ≤ P sup τ ∈ D βn,M sup ( α,β ) ∈ Θ | M βn ( τ : β | α ) | hϑ β ([ nτ ] − [ nτ β ∗ ]) | ˆ β − ˆ β | ≥ δ, | ˆ β − ˆ β | ≤ ϑ β + P ( | ˆ β − ˆ β | > ϑ β ) + P (ˆ α / ∈ O α ∗ ) + P ( ˆ β / ∈ O β ) + P ( ˆ β / ∈ O β ) ≤ P sup τ ∈ D βn,M ( α,β ) ∈ Θ | M βn ( τ : β | α ) | hϑ β ([ nτ ] − [ nτ β ∗ ]) ≥ δ + P ( | ˆ β − ˆ β | > ϑ β ) + P (ˆ α / ∈ O α ∗ ) + P ( ˆ β / ∈ O β ) + P ( ˆ β / ∈ O β ) . (7.45)By the uniform version on the H´ajek-Renyi inequality in Lemma 2 of Iacus and Yoshida (2012), we obtain P sup τ ∈ D βn,M ( α,β ) ∈ Θ | M βn ( τ : β | α ) | hϑ β ([ nτ ] − [ nτ β ∗ ]) ≥ δ = P sup τ ∈ D βn,M ( α,β ) ∈ Θ | M βn ( τ : β | α ) | [ nτ ] − [ nτ β ∗ ] ≥ δhϑ β ≤ P max j>M/hϑ β − j sup ( α,β ) ∈ Θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ β ∗ ]+ j (cid:88) i =[ nτ β ∗ ]+1 (cid:0) ∂ β G i ( β | α ) − E [ ∂ β G i ( β | α ) | G ni − ] (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ δhϑ α ≤ (cid:88) j>M/hϑ β − Ch ( δhϑ β j ) ≤ C (cid:48) h ( δhϑ β ) hϑ β M = C (cid:48) δ M =: γ β ( M ) . (7.46)Noting that if | ˆ β k − β ∗ k | ≤ ϑ β / | ˆ β − ˆ β | ≤ | ˆ β − β ∗ | + | β ∗ − β ∗ | + | ˆ β − β ∗ | ≤ ϑ β , i.e., {| ˆ β − ˆ β | > ϑ β } ⊂ (cid:26) | ˆ β − β ∗ | > ϑ β (cid:27) ∪ (cid:26) | ˆ β − β ∗ | > ϑ β (cid:27) , and for sufficiently large n so that P ( | ˆ β k − β ∗ k | > ϑ β / < (cid:15)/ ϑ − β ( ˆ β k − β ∗ k ) = o p (1), we see P ( | ˆ β − ˆ β | > ϑ β ) ≤ P (cid:18) | ˆ β − β ∗ | > ϑ β (cid:19) + P (cid:18) | ˆ β − β ∗ | > ϑ β (cid:19) < (cid:15). (7.47)From [C6-II] and [A2-II] , we have ˆ α p −→ α ∗ and ˆ β k p −→ β as n → ∞ , that is, P (ˆ α / ∈ O α ∗ ) < (cid:15) and P ( ˆ β k / ∈ O β ) < (cid:15) for large n . Therefore, from (7.45)-(7.47) and these, we have P β ,n ≤ γ β ( M ) + 4 (cid:15) for large n . STIMATION FOR CHANGE POINT OF ERGODIC DIFFUSION PROCESSES 45 [ii] Evaluation of P β ,n . If ˆ α ∈ O α ∗ and ˆ β k ∈ O β , then we havetr (cid:16) A − i − (ˆ α )( b i − ( ˆ β ) − b i − ( ˆ β )) ⊗ (cid:17) = tr (cid:16) A − i − ( α ∗ )( b i − ( ˆ β ) − b i − ( ˆ β )) ⊗ (cid:17) + (cid:90) ∂ α tr (cid:16) A − i − ( α )( b i − ( ˆ β ) − b i − ( ˆ β )) ⊗ (cid:17)(cid:12)(cid:12)(cid:12) α = α ∗ + u (ˆ α − α ∗ ) d u (ˆ α − α ∗ )= ∂ β tr (cid:16) A − i − ( α ∗ )( b i − ( β ) − b i − ( ˆ β )) ⊗ (cid:17)(cid:12)(cid:12)(cid:12) β = ˆ β ( ˆ β − ˆ β )+ 12 ∂ β tr (cid:16) A − i − ( α ∗ )( b i − ( β ) − b i − ( ˆ β )) ⊗ (cid:17)(cid:12)(cid:12)(cid:12) β = ˆ β ⊗ ( ˆ β − ˆ β ) ⊗ + 13! ∂ β tr (cid:16) A − i − ( α ∗ )( b i − ( β ) − b i − ( ˆ β )) ⊗ (cid:17)(cid:12)(cid:12)(cid:12) β = ˆ β ⊗ ( ˆ β − ˆ β ) ⊗ + (cid:90) (1 − u ) ∂ β tr (cid:16) A − i − ( α ∗ )( b i − ( β ) − b i − ( ˆ β )) ⊗ (cid:17)(cid:12)(cid:12)(cid:12) β = ˆ β + u ( ˆ β − ˆ β ) d u ⊗ ( ˆ β − ˆ β ) ⊗ + (cid:90) (cid:90) ∂ β ∂ α tr (cid:16) A − i − ( α )( b i − ( β ) − b i − ( ˆ β )) ⊗ (cid:17)(cid:12)(cid:12)(cid:12) α = α ∗ + u (ˆ α − α ∗ ) β = ˆ β + v ( ˆ β − ˆ β ) d u d v ⊗ (ˆ α − α ∗ ) ⊗ ( ˆ β − ˆ β )= Ξ βi − ( α ∗ , ˆ β ) ⊗ ( ˆ β − ˆ β ) ⊗ + 13! ∂ β tr (cid:16) A − i − ( α ∗ )( b i − ( β ) − b i − ( ˆ β )) ⊗ (cid:17)(cid:12)(cid:12)(cid:12) β = ˆ β ⊗ ( ˆ β − ˆ β ) ⊗ + (cid:90) (1 − u ) ∂ β tr (cid:16) A − i − ( α ∗ )( b i − ( β ) − b i − ( ˆ β )) ⊗ (cid:17)(cid:12)(cid:12)(cid:12) β = ˆ β + u ( ˆ β − ˆ β ) d u ⊗ ( ˆ β − ˆ β ) ⊗ + (cid:90) (cid:90) ∂ β ∂ α tr (cid:16) A − i − ( α )( b i − ( β ) − b i − ( ˆ β )) ⊗ (cid:17)(cid:12)(cid:12)(cid:12) α = α ∗ + u (ˆ α − α ∗ ) β = ˆ β + v ( ˆ β − ˆ β ) d u d v ⊗ (ˆ α − α ∗ ) ⊗ ( ˆ β − ˆ β )= Ξ βi − ( α ∗ , β ) ⊗ ( ˆ β − ˆ β ) ⊗ + ∂ β Ξ βi − ( α ∗ , β ) ⊗ ( ˆ β − ˆ β ) ⊗ ⊗ ( ˆ β − β )+ (cid:90) (1 − u ) ∂ β Ξ βi − ( α ∗ , β + u ( ˆ β − β ))d u ⊗ ( ˆ β − ˆ β ) ⊗ ⊗ ( ˆ β − β ) ⊗ + 13! ∂ β tr (cid:16) A − i − ( α ∗ )( b i − ( β ) − b i − ( ˆ β )) ⊗ (cid:17)(cid:12)(cid:12)(cid:12) β = ˆ β ⊗ ( ˆ β − ˆ β ) ⊗ + (cid:90) (1 − u ) ∂ β tr (cid:16) A − i − ( α ∗ )( b i − ( β ) − b i − ( ˆ β )) ⊗ (cid:17)(cid:12)(cid:12)(cid:12) β = ˆ β + u ( ˆ β − ˆ β ) d u ⊗ ( ˆ β − ˆ β ) ⊗ + (cid:90) (cid:90) ∂ β ∂ α tr (cid:16) A − i − ( α )( b i − ( β ) − b i − ( ˆ β )) ⊗ (cid:17)(cid:12)(cid:12)(cid:12) α = α ∗ + u (ˆ α − α ∗ ) β = ˆ β + v ( ˆ β − ˆ β ) d u d v ⊗ (ˆ α − α ∗ ) ⊗ ( ˆ β − ˆ β ) ≥ (cid:16) λ [Ξ βi − ( α ∗ , β )] + r i − (cid:17) | ˆ β − ˆ β | , where r i − satisfies sup τ ∈ D βn,M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) nτ ] − [ nτ β ∗ ] [ nτ ] (cid:88) i =[ nτ β ∗ ]+1 r i − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = o p (1)from [A1-II] and [A2-II] . Therefore we have P β ,n = P (cid:32) inf τ ∈ D βn,M A βn ( τ : ˆ β , ˆ β | ˆ α ) hϑ β ([ nτ ] − [ nτ β ∗ ]) ≤ δ (cid:33) ≤ P (cid:32) inf τ ∈ D βn,M A βn ( τ : ˆ β , ˆ β | ˆ α ) hϑ β ([ nτ ] − [ nτ β ∗ ]) ≤ δ, | ˆ β − ˆ β | ≥ ϑ β , ˆ α ∈ O α ∗ , ˆ β k ∈ O β (cid:33) + P (cid:18) | ˆ β − ˆ β | < ϑ β (cid:19) + P (ˆ α / ∈ O α ∗ ) + P ( ˆ β / ∈ O β ) + P ( ˆ β / ∈ O β ) ≤ P inf τ ∈ D βn,M ϑ β ([ nτ ] − [ nτ β ∗ ]) [ nτ ] (cid:88) i =[ nτ β ∗ ]+1 (cid:16) λ [Ξ βi − ( α ∗ , β )] + r i − (cid:17) | ˆ β − ˆ β | ≤ δ, | ˆ β − ˆ β | ≥ ϑ β + P (cid:18) | ˆ β − ˆ β | < ϑ β (cid:19) + P (ˆ α / ∈ O α ∗ ) + P ( ˆ β / ∈ O β ) + P ( ˆ β / ∈ O β ) ≤ P inf τ ∈ D βn,M nτ ] − [ nτ β ∗ ] [ nτ ] (cid:88) i =[ nτ β ∗ ]+1 (cid:16) λ [Ξ βi − ( β )] + r i − (cid:17) ≤ δ + P (cid:18) | ˆ β − ˆ β | < ϑ β (cid:19) + P (ˆ α / ∈ O α ∗ ) + P ( ˆ β / ∈ O β ) + P ( ˆ β / ∈ O β ) . (7.48)According to [A2-II] , if we set δ = 110 (cid:90) R d λ [Ξ β ( x, α ∗ , β )]d µ ( α ∗ ,β ) ( x ) > , then for large n , P inf τ ∈ D βn,M nτ ] − [ nτ β ∗ ] [ nτ ] (cid:88) i =[ nτ β ∗ ]+1 (cid:16) λ [Ξ βi − ( α ∗ , β )] + r i − (cid:17) ≤ δ ≤ P inf τ ∈ D βn,M nτ ] − [ nτ β ∗ ] [ nτ ] (cid:88) i =[ nτ β ∗ ]+1 λ [Ξ βi − ( α ∗ , β )] ≤ δ + P inf τ ∈ D βn,M nτ ] − [ nτ β ∗ ] [ nτ ] (cid:88) i =[ nτ β ∗ ]+1 r i − ≤ − δ ≤ P inf τ ∈ D βn,M nτ ] − [ nτ β ∗ ] [ nτ ] (cid:88) i =[ nτ β ∗ ]+1 λ [Ξ βi − ( α ∗ , β )] ≤ δ + P sup τ ∈ D βn,M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) nτ ] − [ nτ β ∗ ] [ nτ ] (cid:88) i =[ nτ β ∗ ]+1 r i − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ δ ≤ P inf τ ∈ D βn,M nτ ] − [ nτ β ∗ ] [ nτ ] (cid:88) i =[ nτ β ∗ ]+1 λ [Ξ βi − ( α ∗ , β )] − δ ≤ − δ + (cid:15) ≤ P sup τ ∈ D βn,M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) nτ ] − [ nτ β ∗ ] [ nτ ] (cid:88) i =[ nτ β ∗ ]+1 λ [Ξ βi − ( α ∗ , β )] − δ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ δ + (cid:15) ≤ P sup k>M/hϑ β − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k [ nτ β ∗ ]+ k (cid:88) i =[ nτ β ∗ ]+1 λ [Ξ βi − ( α ∗ , β )] − δ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ δ + (cid:15) ≤ P max [ n /r ] ≤ k ≤ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k [ nτ β ∗ ]+ k (cid:88) i =[ nτ β ∗ ]+1 λ [Ξ βi − ( α ∗ , β )] − δ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ δ + (cid:15) ≤ (cid:15). (7.49)Noting that if | ˆ β k − β ∗ k | ≤ ϑ β / ϑ β = | β ∗ − β ∗ | ≤ | β ∗ − ˆ β | + | ˆ β − ˆ β | + | ˆ β − β ∗ | ≤ | ˆ β − ˆ β | + ϑ β / (cid:26) | ˆ β − ˆ β | < ϑ β (cid:27) ⊂ (cid:26) | ˆ β − β ∗ | > ϑ β (cid:27) ∪ (cid:26) | ˆ β − β ∗ | > ϑ β (cid:27) and for sufficiently large n so that P ( | ˆ β k − β ∗ k | > ϑ β / < (cid:15)/ ϑ − β ( ˆ β k − β ∗ k ) = o p (1), we see P (cid:18) | ˆ β − ˆ β | < ϑ β (cid:19) ≤ P (cid:18) | ˆ β − β ∗ | > ϑ β (cid:19) + P (cid:18) | ˆ β − β ∗ | > ϑ β (cid:19) < (cid:15). (7.50)Therefore, from (7.48)-(7.50), we obtain P β ,n ≤ (cid:15) for large n . STIMATION FOR CHANGE POINT OF ERGODIC DIFFUSION PROCESSES 47 [iii] Evaluation of P β ,n . We have, for large n ,tr (cid:16) A − i − (ˆ α )( b i − ( ˆ β ) − h E [∆ X i | G ni − ])( b i − ( ˆ β ) − b i − ( ˆ β )) T (cid:17) ≤ h tr (cid:16) A − i − (ˆ α )( b i − ( ˆ β ) − b i − ( β ∗ ))( b i − ( ˆ β ) − b i − ( ˆ β )) T (cid:17) + R i − ( h , θ ) | b i − ( ˆ β ) − b i − ( ˆ β ) | = h (cid:104) tr (cid:16) A − i − (ˆ α ) ∂ β (cid:96) b i − ( β ∗ )( b i − ( ˆ β ) − b i − ( ˆ β )) T (cid:17)(cid:105) (cid:96) ( ˆ β − β ∗ )+ h (cid:90) (1 − u ) (cid:104) tr (cid:16) A − i − (ˆ α ) ∂ β (cid:96) ∂ β (cid:96) b i − ( β )( b i − ( ˆ β ) − b i − ( ˆ β )) T (cid:17)(cid:105) (cid:96) ,(cid:96) (cid:12)(cid:12)(cid:12)(cid:12) β = β ∗ + u ( ˆ β − β ∗ ) d u ⊗ ( ˆ β − β ∗ ) ⊗ + R i − ( h , θ ) | b i − ( ˆ β ) − b i − ( ˆ β ) | = h (cid:104) tr (cid:16) A − i − (ˆ α ) ∂ β (cid:96) b i − ( β ∗ ) ∂ β (cid:96) b i − ( ˆ β ) T (cid:17)(cid:105) (cid:96) ,(cid:96) ⊗ ( ˆ β − β ∗ ) ⊗ ( ˆ β − ˆ β )+ h (cid:90) (1 − u ) (cid:104) tr (cid:16) A − i − (ˆ α ) ∂ β (cid:96) b i − ( β ∗ ) ∂ β (cid:96) ∂ β (cid:96) b i − ( β ) T (cid:17)(cid:105) (cid:96) ,(cid:96) ,(cid:96) (cid:12)(cid:12)(cid:12)(cid:12) β = ˆ β + u ( ˆ β − ˆ β ) d u ⊗ ( ˆ β − β ∗ ) ⊗ ( ˆ β − ˆ β ) ⊗ + h (cid:90) (1 − u ) (cid:104) tr (cid:16) A − i − (ˆ α ) ∂ β (cid:96) ∂ β (cid:96) b i − ( β )( b i − ( ˆ β ) − b i − ( ˆ β )) T (cid:17)(cid:105) (cid:96) ,(cid:96) (cid:12)(cid:12)(cid:12)(cid:12) β = β ∗ + u ( ˆ β − β ∗ ) d u ⊗ ( ˆ β − β ∗ ) ⊗ + R i − ( h , θ ) | b i − ( ˆ β ) − b i − ( ˆ β ) | = h Ξ βi − (ˆ α, β ∗ ) ⊗ ( ˆ β − β ∗ ) ⊗ ( ˆ β − ˆ β )+ h (cid:90) (cid:104) tr (cid:16) A − i − (ˆ α ) ∂ β (cid:96) b i − ( β ∗ ) ∂ β (cid:96) ∂ β (cid:96) b i − ( β ∗ + u ( ˆ β − β ∗ )) T (cid:17)(cid:105) (cid:96) ,(cid:96) ,(cid:96) d u ⊗ ( ˆ β − β ∗ ) ⊗ ⊗ ( ˆ β − ˆ β )+ h (cid:90) (1 − u ) (cid:104) tr (cid:16) A − i − (ˆ α ) ∂ β (cid:96) b i − ( β ∗ ) ∂ β (cid:96) ∂ β (cid:96) b i − ( ˆ β + u ( ˆ β − ˆ β )) T (cid:17)(cid:105) (cid:96) ,(cid:96) ,(cid:96) d u ⊗ ( ˆ β − β ∗ ) ⊗ ( ˆ β − ˆ β ) ⊗ + h (cid:90) (1 − u ) (cid:104) tr (cid:16) A − i − (ˆ α ) ∂ β (cid:96) ∂ β (cid:96) b i − ( β ∗ + u ( ˆ β − β ∗ ))( b i − ( ˆ β ) − b i − ( ˆ β )) T (cid:17)(cid:105) (cid:96) ,(cid:96) d u ⊗ ( ˆ β − β ∗ ) ⊗ + R i − ( h , θ ) | b i − ( ˆ β ) − b i − ( ˆ β ) | = h Ξ βi − ( α ∗ , β ) ⊗ ( ˆ β − β ∗ ) ⊗ ( ˆ β − ˆ β )+ h (cid:90) ∂ ( α,β ) Ξ βi − ( α, β ) (cid:12)(cid:12)(cid:12) α = α ∗ + u (ˆ α − α ∗ ) β = β + u ( β ∗ − β ) d u ⊗ ( ˆ β − β ∗ ) ⊗ ( ˆ β − ˆ β ) ⊗ (cid:18) ˆ α − α ∗ β ∗ − β (cid:19) + h (cid:90) (cid:104) tr (cid:16) A − i − (ˆ α ) ∂ β (cid:96) b i − ( β ∗ ) ∂ β (cid:96) ∂ β (cid:96) b i − ( β ∗ + u ( ˆ β − β ∗ )) T (cid:17)(cid:105) (cid:96) ,(cid:96) ,(cid:96) d u ⊗ ( ˆ β − β ∗ ) ⊗ ⊗ ( ˆ β − ˆ β )+ h (cid:90) (1 − u ) (cid:104) tr (cid:16) A − i − (ˆ α ) ∂ β (cid:96) b i − ( β ∗ ) ∂ β (cid:96) ∂ β (cid:96) b i − ( ˆ β + u ( ˆ β − ˆ β )) T (cid:17)(cid:105) (cid:96) ,(cid:96) ,(cid:96) d u ⊗ ( ˆ β − β ∗ ) ⊗ ( ˆ β − ˆ β ) ⊗ + h (cid:90) (1 − u ) (cid:104) tr (cid:16) A − i − (ˆ α ) ∂ β (cid:96) ∂ β (cid:96) b i − ( β ∗ + u ( ˆ β − β ∗ ))( b i − ( ˆ β ) − b i − ( ˆ β )) T (cid:17)(cid:105) (cid:96) ,(cid:96) d u ⊗ ( ˆ β − β ∗ ) ⊗ + R i − ( h , θ ) | b i − ( ˆ β ) − b i − ( ˆ β ) |≤ h Ξ βi − ( α ∗ , β ) ⊗ ( ˆ β − β ∗ ) ⊗ ( ˆ β − ˆ β )+ hϑ β √ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∂ ( α,β ) Ξ βi − ( α, β ) (cid:12)(cid:12)(cid:12) α = α ∗ + u (ˆ α − α ∗ ) β = β + u ( β ∗ − β ) d u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ T | ˆ β − β ∗ | ϑ − β | ˆ β − ˆ β | ( | ˆ α − α ∗ | + | β ∗ − β | ) + hϑ β T (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) (cid:104) tr (cid:16) A − i − (ˆ α ) ∂ β (cid:96) b i − ( β ∗ ) ∂ β (cid:96) ∂ β (cid:96) b i − ( β ∗ + u ( ˆ β − β ∗ )) T (cid:17)(cid:105) (cid:96) ,(cid:96) ,(cid:96) d u (cid:12)(cid:12)(cid:12)(cid:12) × (cid:16) √ T | ˆ β − β ∗ | (cid:17) ϑ − β | ˆ β − ˆ β | + hϑ β √ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) (1 − u ) (cid:104) tr (cid:16) A − i − (ˆ α ) ∂ β (cid:96) b i − ( β ∗ ) ∂ β (cid:96) ∂ β (cid:96) b i − ( ˆ β + u ( ˆ β − ˆ β )) T (cid:17)(cid:105) (cid:96) ,(cid:96) ,(cid:96) d u (cid:12)(cid:12)(cid:12)(cid:12) × √ T | ˆ β − β ∗ | (cid:16) ϑ − β | ˆ β − ˆ β | (cid:17) + hϑ β T (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) (cid:90) (1 − u ) (cid:104) tr (cid:16) A − i − (ˆ α ) ∂ β (cid:96) ∂ β (cid:96) b i − ( β ∗ + u ( ˆ β − β ∗ )) × ∂ β (cid:96) b i − ( ˆ β + v ( ˆ β − ˆ β )) T (cid:17)(cid:105) (cid:96) ,(cid:96) ,(cid:96) d u d v (cid:12)(cid:12)(cid:12)(cid:12) (cid:16) √ T | ˆ β − β ∗ | (cid:17) ϑ − β | ˆ β − ˆ β | + R i − ( h , θ ) | b i − ( ˆ β ) − b i − ( ˆ β ) |≤ hϑ β √ T Ξ βi − ( α ∗ , β ) ⊗ √ T ( ˆ β − β ∗ ) ⊗ ϑ − β ( ˆ β − ˆ β ) + (cid:32) hϑ β √ nT + hϑ β √ T + hϑ β T + h (cid:33) R i − (1 , θ )andsup τ ∈ D βn,M | (cid:37) βn ( τ : ˆ β , ˆ β | ˆ α ) | hϑ β ([ nτ ] − [ nτ β ∗ ]) ≤ √ T ϑ β sup τ ∈ D βn,M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) nτ ] − [ nτ β ∗ ] [ nτ ] (cid:88) i =[ nτ β ∗ ]+1 Ξ βi − ( α ∗ , β ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ T | ˆ β − β ∗ | ϑ − β | ˆ β − ˆ β | + (cid:32) hϑ β √ nT + hϑ β √ T + hϑ β T + h (cid:33) sup τ ∈ D βn,M hϑ β ([ nτ ] − [ nτ β ∗ ]) [ nτ ] (cid:88) i =[ nτ β ∗ ]+1 R i − (1 , θ ) ≤ O p (cid:32) √ T ϑ β (cid:33) + O p ( √ hϑ β ) + O p ( √ T ϑ β ) + O p ( ϑ β ) + O p ( nh ) = o p (1) . Hence, we see P β ,n ≤ P sup τ ∈ D βn,M | (cid:37) βn ( τ : ˆ β , ˆ β | ˆ α ) | hϑ β ([ nτ ] − [ nτ β ∗ ]) ≥ δ, ˆ α ∈ O α ∗ , ˆ β , ˆ β ∈ O β + P (ˆ α / ∈ O α ∗ ) + P ( ˆ β / ∈ O β ) + P ( ˆ β / ∈ O β ) ≤ (cid:15) for large n .[iv] From the evaluations in Steps [i]-[iii], we havelim n →∞ P ( T ϑ β (ˆ τ βn − τ β ∗ ) > M ) ≤ γ β ( M ) + 14 (cid:15) for any M ≥ (cid:15) >
0. Thereforelim M →∞ lim n →∞ P ( T ϑ β (ˆ τ βn − τ β ∗ ) > M ) ≤ (cid:15). (cid:3) Proof of Theorem 4.
It is sufficient to showlim M →∞ lim n →∞ P ( T (ˆ τ βn − τ β ∗ ) > M ) = 0 . Let M ≥ D βn,M = { τ ∈ [0 , | T ( τ − τ β ∗ ) > M } . For all δ >
0, we have P (cid:0) T (ˆ τ βn − τ β ∗ ) > M (cid:1) ≤ P sup τ ∈ D βn,M |M βn ( τ : ˆ β , ˆ β | ˆ α ) | h ([ nτ ] − [ nτ β ∗ ]) ≥ δ + P (cid:32) inf τ ∈ D βn,M A βn ( τ : ˆ β , ˆ β | ˆ α ) h ([ nτ ] − [ nτ β ∗ ]) ≤ δ (cid:33) STIMATION FOR CHANGE POINT OF ERGODIC DIFFUSION PROCESSES 49 + P sup τ ∈ D βn,M | (cid:37) βn ( τ : ˆ β , ˆ β | ˆ α ) | h ([ nτ ] − [ nτ β ∗ ]) ≥ δ =: P β ,n + P β ,n + P β ,n . [i] Evaluation of P β ,n . For large n , we have P β ,n ≤ P sup τ ∈ D βn,M |M βn ( τ : ˆ β , ˆ β | ˆ α ) | h ([ nτ ] − [ nτ β ∗ ]) ≥ δ, ˆ α ∈ O α ∗ . ˆ β ∈ O β ∗ , ˆ β ∈ O β ∗ + P ( ˆ β ∈ O β ∗ ) + P ( ˆ β ∈ O β ∗ ) + P (ˆ α ∈ O α ∗ ) ≤ P sup τ ∈ D βn,M sup α ∈O α ∗ ,β k ∈O β ∗ k |M βn ( τ : β , β | α ) | [ nτ ] − [ nτ β ∗ ] ≥ δh + P ( ˆ β ∈ O β ∗ ) + P ( ˆ β ∈ O β ∗ ) + P (ˆ α ∈ O α ∗ )By the uniform version on the H´ajek-Renyi inequality in Lemma 2 of Iacus and Yoshida (2012), we obtain P sup τ ∈ D βn,M nτ ] − [ nτ β ∗ ] sup α ∈O α ∗ ,β k ∈O β ∗ k |M βn ( τ : β , β | α ) | ≥ δh ≤ P max j>M/h − j sup α ∈O α ∗ ,β k ∈O β ∗ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ β ∗ ]+ j (cid:88) i =[ nτ β ∗ ]+1 (cid:0) G i ( β | α ) − G i ( β | α ) − E β ∗ [ G i ( β | α ) − G i ( β | α ) | G ni − ] (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ δh ≤ (cid:88) j>M/h − Ch ( δhj ) ≤ C (cid:48) h ( δh ) hM = C (cid:48) δ M =: γ β ( M ) . From [C6-II] , we have P (ˆ α / ∈ O α ∗ ) < (cid:15) and P ( ˆ β k / ∈ O β ∗ k ) < (cid:15) for large n . Therefore P β ,n ≤ γ β ( M ) + 3 (cid:15) forlarge n .[ii] Evaluation of P β ,n . If ˆ α ∈ O α ∗ and ˆ β k ∈ O β ∗ k , then there exists a positive constant c independent of i such thatΓ βi − (ˆ α, ˆ β , ˆ β ) = Γ βi − ( α ∗ , β ∗ , β ∗ ) + (cid:90) ∂ ( α,β ,β ) Γ βi − ( α, β , β ) (cid:12)(cid:12)(cid:12) α = α ∗ + u (ˆ α − α ∗ ) β k = β ∗ k + u ( ˆ β k − β ∗ k ) d u ˆ α − α ∗ ˆ β − β ∗ ˆ β − β ∗ ≥ Γ βi − ( α ∗ , β ∗ , β ∗ ) − c ( | ˆ α − α ∗ | + | ˆ β − β ∗ | + | ˆ β − β ∗ | ) . According to [B1-II] , if we set δ = 14 inf x Γ β ( x, α ∗ , β ∗ , β ∗ ) > , then for large n , P β ,n ≤ P (cid:32) inf τ ∈ D βn,M A βn ( τ : ˆ β , ˆ β | ˆ α ) h ([ nτ ] − [ nτ β ∗ ]) ≤ δ, ˆ α ∈ O α ∗ , ˆ β ∈ O β ∗ ˆ β ∈ O β ∗ (cid:33) + P (ˆ α / ∈ O α ∗ ) + P ( ˆ β / ∈ O β ∗ ) + P ( ˆ β / ∈ O β ∗ ) ≤ P inf τ ∈ D βn,M nτ ] − [ nτ β ∗ ] [ nτ ] (cid:88) i =[ nτ β ∗ ]+1 (cid:16) Γ βi − ( α ∗ , β ∗ , β ∗ ) − c (cid:0) | ˆ α − α ∗ | + | ˆ β − β ∗ | + | ˆ β − β ∗ | (cid:1)(cid:17) ≤ δ + P (ˆ α / ∈ O α ∗ ) + P ( ˆ β / ∈ O β ∗ ) + P ( ˆ β / ∈ O β ∗ ) ≤ P inf τ ∈ D βn,M nτ ] − [ nτ β ∗ ] [ nτ ] (cid:88) i =[ nτ β ∗ ]+1 Γ βi − ( α ∗ , β ∗ , β ∗ ) ≤ δ + P (cid:16) − c (cid:0) | ˆ α − α ∗ | + | ˆ β − β ∗ | + | ˆ β − β ∗ | (cid:1) ≤ − δ (cid:17) + P (ˆ α / ∈ O α ∗ ) + P ( ˆ β / ∈ O β ∗ ) + P ( ˆ β / ∈ O β ∗ ) ≤ P (cid:16) inf x Γ β ( x, α ∗ , β ∗ , β ∗ ) ≤ δ (cid:17) + P (cid:18) | ˆ α − α ∗ | + | ˆ β − β ∗ | + | ˆ β − β ∗ | ≥ δc (cid:19) + P (ˆ α / ∈ O α ∗ ) + P ( ˆ β / ∈ O β ∗ ) + P ( ˆ β / ∈ O β ∗ ) ≤ (cid:15) thanks to P (cid:18) | ˆ α − α ∗ | + | ˆ β − β ∗ | + | ˆ β − β ∗ | ≥ δc (cid:19) ≤ P (cid:18) | ˆ α − α ∗ | ≥ δ c (cid:19) + P (cid:18) | ˆ β − β ∗ | ≥ δ c (cid:19) + P (cid:18) | ˆ β − β ∗ | ≥ δ c (cid:19) ≤ (cid:15) from [C6-II] .[iii] Evaluation of P β ,n . We have, for large n ,tr (cid:16) A − i − (ˆ α )( b i − ( ˆ β ) − h E β ∗ [∆ X i | G ni − ])( b i − ( ˆ β ) − b i − ( ˆ β )) T (cid:17) = h tr (cid:16) A − i − (ˆ α )( b i − ( ˆ β ) − b i − ( β ∗ ))( b i − ( ˆ β ) − b i − ( ˆ β )) T (cid:17) + R i − ( h , θ )= h (cid:90) ∂ β tr (cid:16) A − i − (ˆ α )( b i − ( β ) − b i − ( β ∗ ))( b i − ( ˆ β ) − b i − ( ˆ β )) T (cid:17)(cid:12)(cid:12)(cid:12) β = β ∗ + u ( ˆ β − β ∗ ) d u ( ˆ β − β ∗ ) + R i − ( h , θ )and sup τ ∈ D βn,M | (cid:37) βn ( τ : ˆ β , ˆ β ) | h ([ nτ ] − [ nτ β ∗ ]) ≤ √ T sup τ ∈ D βn,M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) nτ ] − [ nτ β ∗ ] [ nτ ] (cid:88) i =[ nτ β ∗ ]+1 (cid:90) ∂ β tr (cid:16) A − i − (ˆ α )( b i − ( β ) − b i − ( β ∗ ))( b i − ( ˆ β ) − b i − ( ˆ β )) T (cid:17)(cid:12)(cid:12)(cid:12) β = β ∗ + u ( ˆ β − β ∗ ) d u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ T | ˆ β − β ∗ | + sup τ ∈ D βn,M h h ([ nτ ] − [ nτ β ∗ ]) [ nτ ] (cid:88) i =[ nτ β ∗ ]+1 R i − (1 , θ ) ≤ √ T sup x,α,β k (cid:12)(cid:12)(cid:12)(cid:104) tr (cid:16) A − ( x, α ) ∂ β (cid:96) b ( x, β ) (cid:0) b ( x, β ) − b ( x, β ) (cid:1) T (cid:17)(cid:105) (cid:96) (cid:12)(cid:12)(cid:12) √ T | ˆ β − β ∗ | + h M n (cid:88) i =[ nτ β ∗ ]+1 R i − (1 , θ )= o p (1) . Hence, we see P β ,n ≤ P sup τ ∈ D βn,M | (cid:37) βn ( τ : ˆ β , ˆ β ) | h ([ nτ ] − [ nτ β ∗ ]) ≥ δ, ˆ α ∈ O α ∗ , ˆ β ∈ O β ∗ , ˆ β ∈ O β ∗ + P (ˆ α / ∈ O α ∗ ) + P ( ˆ β / ∈ O β ∗ ) + P ( ˆ β / ∈ O β ∗ ) ≤ (cid:15) for large n . STIMATION FOR CHANGE POINT OF ERGODIC DIFFUSION PROCESSES 51 [iv] From the evaluations in Steps [i]-[iii], we havelim n →∞ P ( T (ˆ τ βn − τ β ∗ ) > M ) ≤ γ β ( M ) + 13 (cid:15) for any M ≥ (cid:15) >
0. Thereforelim M →∞ lim n →∞ P ( T (ˆ τ βn − τ β ∗ ) > M ) ≤ (cid:15). (cid:3) Proof of Proposition 1.
We see T αn = 1 √ dn max ≤ k ≤ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k (cid:88) i =1 ˆ η i − kn n (cid:88) i =1 ˆ η i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ √ dn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ α ∗ ] (cid:88) i =1 ˆ η i − [ nτ α ∗ ] n n (cid:88) i =1 ˆ η i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:114) nϑ α d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) nϑ α [ nτ α ∗ ] (cid:88) i =1 ˆ η i − [ nτ α ∗ ] n nϑ α n (cid:88) i =1 ˆ η i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (7.51)Now we can expressˆ η i = tr (cid:18) A − i − (ˆ α ) (∆ X i ) ⊗ h (cid:19) = tr (cid:18) A − i − ( α ) (∆ X i ) ⊗ h (cid:19) + ∂ α tr (cid:18) A − i − ( α ) (∆ X i ) ⊗ h (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) α = α (ˆ α − α )+ (cid:90) (1 − u ) ∂ α tr (cid:18) A − i − ( α ) (∆ X i ) ⊗ h (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) α = α + u (ˆ α − α ) d u ⊗ (ˆ α − α ) ⊗ = tr (cid:18) A − i − ( α ) (∆ X i ) ⊗ h (cid:19) + (cid:18) tr (cid:18) A − ∂ α (cid:96) AA − i − ( α ) (∆ X i ) ⊗ h (cid:19)(cid:19) (cid:96) (ˆ α − α )+ (cid:90) (1 − u ) ∂ α tr (cid:18) A − i − ( α ) (∆ X i ) ⊗ h (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) α = α + u (ˆ α − α ) d u ⊗ (ˆ α − α ) ⊗ =: η ,i + η ,i (ˆ α − α ) + (cid:90) (1 − u ) ∂ α tr (cid:18) A − i − ( α ) (∆ X i ) ⊗ h (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) α = α + u (ˆ α − α ) d u ⊗ (ˆ α − α ) ⊗ . Therefore we have, from [E1] ,1 nϑ α [ nτ α ∗ ] (cid:88) i =1 ˆ η i − [ nτ α ∗ ] n nϑ α n (cid:88) i =1 ˆ η i = 1 nϑ α [ nτ α ∗ ] (cid:88) i =1 η ,i − [ nτ α ∗ ] n nϑ α n (cid:88) i =1 η ,i + n [ nτ α ∗ ] (cid:88) i =1 η ,i − [ nτ α ∗ ] n n n (cid:88) i =1 η ,i ϑ − α (ˆ α − α ) + o p (1)=: H ,n + H ,n ϑ − α (ˆ α − α ) + o p (1) . (7.52)Here H ,n and H ,n can be transformed as follows. H ,n = (cid:18) − [ nτ α ∗ ] n (cid:19) nϑ α [ nτ α ∗ ] (cid:88) i =1 η ,i − [ nτ α ∗ ] n nϑ α n (cid:88) i =[ nτ α ∗ ]+1 η ,i , H ,n = (cid:18) − [ nτ α ∗ ] n (cid:19) n [ nτ α ∗ ] (cid:88) i =1 η ,i − [ nτ α ∗ ] n n n (cid:88) i =[ nτ α ∗ ]+1 η ,i . We have (cid:18) − [ nτ α ∗ ] n (cid:19) nϑ α [ nτ α ∗ ] (cid:88) i =1 E α ∗ [ η ,i | G ni − ] − [ nτ α ∗ ] n nϑ α n (cid:88) i =[ nτ α ∗ ]+1 E α ∗ [ η ,i | G ni − ] = (cid:18) − [ nτ α ∗ ] n (cid:19) nϑ α [ nτ α ∗ ] (cid:88) i =1 tr( A − i − ( α ) A i − ( α ∗ )) − [ nτ α ∗ ] n nϑ α n (cid:88) i =[ nτ α ∗ ]+1 tr( A − i − ( α ) A i − ( α ∗ )) + O p ( h/ϑ α )= (cid:18) − [ nτ α ∗ ] n (cid:19) nϑ α [ nτ α ∗ ] (cid:88) i =1 (cid:18) d + ∂ α tr( A − i − ( α ) A i − ( α )) (cid:12)(cid:12) α = α ( α ∗ − α )+ (cid:90) (1 − u ) ∂ α tr( A − i − ( α ) A i − ( α )) (cid:12)(cid:12) α = α + u ( α ∗ − α ) ⊗ ( α ∗ − α ) ⊗ (cid:19) − [ nτ α ∗ ] n nϑ α n (cid:88) i =[ nτ α ∗ ]+1 (cid:18) d + ∂ α tr( A − i − ( α ) A i − ( α )) (cid:12)(cid:12) α = α ( α ∗ − α )+ (cid:90) (1 − u ) ∂ α tr( A − i − ( α ) A i − ( α )) (cid:12)(cid:12) α = α + u ( α ∗ − α ) ⊗ ( α ∗ − α ) ⊗ (cid:19) + o p (1)= (cid:18) − [ nτ α ∗ ] n (cid:19) n [ nτ α ∗ ] (cid:88) i =1 (cid:16) tr (cid:0) A − i − ( α ) ∂ α (cid:96) A i − ( α ) (cid:1)(cid:17) (cid:96) ϑ − α ( α ∗ − α ) − [ nτ α ∗ ] n n n (cid:88) i =[ nτ α ∗ ]+1 (cid:16) tr (cid:0) A − i − ( α ) ∂ α (cid:96) A i − ( α ) (cid:1)(cid:17) (cid:96) ϑ − α ( α ∗ − α ) + o p (1) p −→ τ α ∗ (1 − τ α ∗ ) (cid:90) R d (cid:16) tr (cid:0) A − ∂ α (cid:96) A ( x, α ) (cid:1)(cid:17) (cid:96) d µ α ( x )( c − c )and (cid:18) − [ nτ α ∗ ] n (cid:19) nϑ α ) nτ α ∗ ] (cid:88) i =1 E α ∗ [ η ,i | G ni − ] − (cid:18) [ nτ α ∗ ] n (cid:19) nϑ α ) n (cid:88) i =[ nτ α ∗ ]+1 E α ∗ [ η ,i | G ni − ]= 1 nϑ α (cid:18) − [ nτ α ∗ ] n (cid:19) n [ nτ α ∗ ] (cid:88) i =1 E α ∗ [ η ,i | G ni − ] − (cid:18) [ nτ α ∗ ] n (cid:19) n n (cid:88) i =[ nτ α ∗ ]+1 E α ∗ [ η ,i | G ni − ] p −→ . Therefore, from Lemma 9 of Genon-Catalot and Jacod (1993), we obtain H ,n p −→ τ α ∗ (1 − τ α ∗ ) (cid:90) R d (cid:16) tr (cid:0) A − ∂ α (cid:96) A ( x, α ) (cid:1)(cid:17) (cid:96) d µ α ( x )( c − c ) . (7.53)Further, we have (cid:18) − [ nτ α ∗ ] n (cid:19) n [ nτ α ∗ ] (cid:88) i =1 E α ∗ [ η ,i | G ni − ] − [ nτ α ∗ ] n n n (cid:88) i =[ nτ α ∗ ]+1 E α ∗ [ η ,i | G ni − ]= (cid:18) − [ nτ α ∗ ] n (cid:19) n [ nτ α ∗ ] (cid:88) i =1 (cid:104) tr (cid:16) A − ∂ α (cid:96) AA − i − ( α ) A i − ( α ∗ ) (cid:17)(cid:105) (cid:96) − [ nτ α ∗ ] n n n (cid:88) i =[ nτ α ∗ ]+1 (cid:104) tr (cid:16) A − ∂ α (cid:96) AA − i − ( α ) A i − ( α ∗ ) (cid:17)(cid:105) (cid:96) + O p ( h )= (cid:18) − [ nτ α ∗ ] n (cid:19) n [ nτ α ∗ ] (cid:88) i =1 (cid:18)(cid:104) tr (cid:16) A − ∂ α (cid:96) A i − ( α ) (cid:17)(cid:105) (cid:96) + (cid:90) ∂ α (cid:104) tr (cid:16) A − ∂ α (cid:96) AA − i − ( α ) A i − ( α ) (cid:17)(cid:105) (cid:96) (cid:12)(cid:12)(cid:12) α = α + u ( α ∗ − α ) ( α ∗ − α ) (cid:19) STIMATION FOR CHANGE POINT OF ERGODIC DIFFUSION PROCESSES 53 − [ nτ α ∗ ] n n n (cid:88) i =[ nτ α ∗ ]+1 (cid:18)(cid:104) tr (cid:16) A − ∂ α (cid:96) A i − ( α ) (cid:17)(cid:105) (cid:96) + (cid:90) ∂ α (cid:104) tr (cid:16) A − ∂ α (cid:96) AA − i − ( α ) A i − ( α ) (cid:17)(cid:105) (cid:96) (cid:12)(cid:12)(cid:12) α = α + u ( α ∗ − α ) ( α ∗ − α ) (cid:19) + o p (1)= (cid:18) − [ nτ α ∗ ] n (cid:19) n [ nτ α ∗ ] (cid:88) i =1 (cid:104) tr (cid:16) A − ∂ α (cid:96) A i − ( α ) (cid:17)(cid:105) (cid:96) − [ nτ α ∗ ] n n n (cid:88) i =[ nτ α ∗ ]+1 (cid:104) tr (cid:16) A − ∂ α (cid:96) A i − ( α ) (cid:17)(cid:105) (cid:96) + o p (1) p −→ (cid:18) − [ nτ α ∗ ] n (cid:19) n nτ α ∗ ] (cid:88) i =1 E α ∗ [ η ,i | G ni − ] − (cid:18) [ nτ α ∗ ] n (cid:19) n n (cid:88) i =[ nτ α ∗ ]+1 E α ∗ [ η ,i | G ni − ]= 1 n (cid:18) − [ nτ α ∗ ] n (cid:19) n [ nτ α ∗ ] (cid:88) i =1 E α ∗ [ η ,i | G ni − ] − (cid:18) [ nτ α ∗ ] n (cid:19) n n (cid:88) i =[ nτ α ∗ ]+1 E α ∗ [ η ,i | G ni − ] p −→ . Therefore, from Lemma 9 of Genon-Catalot and Jacod (1993), we see H ,n p −→
0. Hence, from (7.52), (7.53)and this, we obtain1 nϑ α [ nτ α ∗ ] (cid:88) i =1 ˆ η i − [ nτ α ∗ ] n nϑ α n (cid:88) i =1 ˆ η i p −→ τ α ∗ (1 − τ α ∗ ) (cid:90) R d (cid:104) tr (cid:16) A − ∂ α (cid:96) A ( x, α ) (cid:17)(cid:105) (cid:96) d µ α ( x )( c − c ) (cid:54) = 0 , which implies T αn −→ ∞ from (7.51) and nϑ α −→ ∞ , that is, P ( T αn > w ( (cid:15) )) −→ (cid:3) Proof of Proposition 2.
We see T β ,n = 1 √ dT max ≤ k ≤ n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k (cid:88) i =1 ˆ ξ i − kn n (cid:88) i =1 ˆ ξ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ √ dT (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) [ nτ β ∗ ] (cid:88) i =1 ˆ ξ i − [ nτ β ∗ ] n n (cid:88) i =1 ˆ ξ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:115) T ϑ β d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ϑ β [ nτ β ∗ ] (cid:88) i =1 ˆ ξ i − [ nτ β ∗ ] n T ϑ β n (cid:88) i =1 ˆ ξ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (7.54)Now we can expressˆ ξ i = 1 T d a − i − (ˆ α )(∆ X i − hb i − ( ˆ β ))= 1 T d a − i − ( α ∗ )(∆ X i − hb i − ( ˆ β )) + (cid:90) ∂ α (cid:16) T d a − i − ( α )(∆ X i − hb i − ( ˆ β )) (cid:17)(cid:12)(cid:12)(cid:12) α = α ∗ + u (ˆ α − α ∗ ) d u (ˆ α − α ∗ )= 1 T d a − i − ( α ∗ )(∆ X i − hb i − ( ¯ β ∗ )) − h T d a − i − ( α ∗ )( b i − ( ˆ β ) − b i − ( ¯ β ∗ ))+ (cid:90) ∂ α (cid:16) T d a − i − ( α )(∆ X i − hb i − ( ˆ β )) (cid:17)(cid:12)(cid:12)(cid:12) α = α ∗ + u (ˆ α − α ∗ ) d u (ˆ α − α ∗ )=: ξ i − h (cid:90) T d a − i − ( α ∗ ) ∂ β b i − ( ¯ β ∗ + u ( ˆ β − ¯ β ∗ ))d u ( ˆ β − ¯ β ∗ )+ (cid:90) ∂ α (cid:16) T d a − i − ( α )(∆ X i − hb i − ( ˆ β )) (cid:17)(cid:12)(cid:12)(cid:12) α = α ∗ + u (ˆ α − α ∗ ) d u (ˆ α − α ∗ )Therefore, we have1 T ϑ β [ nτ β ∗ ] (cid:88) i =1 ˆ ξ i − [ nτ β ∗ ] n T ϑ β n (cid:88) i =1 ˆ ξ i = 1 T ϑ β [ nτ β ∗ ] (cid:88) i =1 ξ i − [ nτ β ∗ ] n T ϑ β n (cid:88) i =1 ξ i + o p (1) = (cid:32) − [ nτ β ∗ ] n (cid:33) T ϑ β [ nτ β ∗ ] (cid:88) i =1 ξ i − [ nτ β ∗ ] n T ϑ β n (cid:88) i =[ nτ β ∗ ]+1 ξ i + o p (1) , (7.55) (cid:32) − [ nτ β ∗ ] n (cid:33) T ϑ β [ nτ β ∗ ] (cid:88) i =1 E β ∗ [ ξ i | G ni − ] − [ nτ β ∗ ] n T ϑ β n (cid:88) i =[ nτ β ∗ ]+1 E β ∗ [ ξ i | G ni − ]= (cid:32) − [ nτ β ∗ ] n (cid:33) T ϑ β [ nτ β ∗ ] (cid:88) i =1 h T d a − i − ( α ∗ )( b i − ( β ∗ ) − b i − ( ¯ β ∗ )) − [ nτ β ∗ ] n T ϑ β n (cid:88) i =[ nτ β ∗ ]+1 h T d a − i − ( α ∗ )( b i − ( β ∗ ) − b i − ( ¯ β ∗ )) + O p ( h/ϑ β )= (cid:32) − [ nτ β ∗ ] n (cid:33) nϑ β [ nτ β ∗ ] (cid:88) i =1 T d a − i − ( α ∗ )( b i − ( β ) − b i − ( ¯ β ∗ )) − [ nτ β ∗ ] n nϑ β n (cid:88) i =[ nτ β ∗ ]+1 T d a − i − ( α ∗ )( b i − ( β ) − b i − ( ¯ β ∗ ))+ (cid:32) − [ nτ β ∗ ] n (cid:33) nϑ β [ nτ β ∗ ] (cid:88) i =1 T d a − i − ( α ∗ ) ∂ β b i − ( β )( β ∗ − β ) − [ nτ β ∗ ] n nϑ β n (cid:88) i =[ nτ β ∗ ]+1 T d a − i − ( α ∗ ) ∂ β b i − ( β )( β ∗ − β )+ (cid:32) − [ nτ β ∗ ] n (cid:33) nϑ β [ nτ β ∗ ] (cid:88) i =1 T d a − i − ( α ∗ ) (cid:90) (1 − u ) ∂ β b i − ( β + u ( β ∗ − β ))d u ⊗ ( β ∗ − β ) ⊗ − [ nτ β ∗ ] n nϑ β n (cid:88) i =[ nτ β ∗ ]+1 T d a − i − ( α ∗ ) (cid:90) (1 − u ) ∂ β b i − ( β + u ( β ∗ − β ))d u ⊗ ( β ∗ − β ) ⊗ + o p (1)= − (cid:32) − [ nτ β ∗ ] n (cid:33) nϑ β [ nτ β ∗ ] (cid:88) i =1 T d a − i − ( α ∗ ) (cid:18) ∂ β b i − ( β )( ¯ β ∗ − β )+ (cid:90) (1 − u ) ∂ β b i − ( β + u ( ¯ β ∗ − β ))d u ⊗ ( ¯ β ∗ − β ) ⊗ (cid:19) + [ nτ β ∗ ] n nϑ β n (cid:88) i =[ nτ β ∗ ]+1 T d a − i − ( α ∗ ) (cid:18) ∂ β b i − ( β )( ¯ β ∗ − β )+ (cid:90) (1 − u ) ∂ β b i − ( β + u ( ¯ β ∗ − β ))d u ⊗ ( ¯ β ∗ − β ) ⊗ (cid:19) + (cid:32) − [ nτ β ∗ ] n (cid:33) n [ nτ β ∗ ] (cid:88) i =1 T d a − i − ( α ∗ ) ∂ β b i − ( β ) ϑ − β ( β ∗ − β ) − [ nτ β ∗ ] n n n (cid:88) i =[ nτ β ∗ ]+1 T d a − i − ( α ∗ ) ∂ β b i − ( β ) ϑ − β ( β ∗ − β ) + o p (1)= − (cid:32) − [ nτ β ∗ ] n (cid:33) n [ nτ β ∗ ] (cid:88) i =1 T d a − i − ( α ∗ ) ∂ β b i − ( β ) ϑ − β ( ¯ β ∗ − β ) STIMATION FOR CHANGE POINT OF ERGODIC DIFFUSION PROCESSES 55 + [ nτ β ∗ ] n n n (cid:88) i =[ nτ β ∗ ]+1 T d a − i − ( α ∗ ) ∂ β b i − ( β ) ϑ − β ( ¯ β ∗ − β )+ (cid:32) − [ nτ β ∗ ] n (cid:33) n [ nτ β ∗ ] (cid:88) i =1 T d a − i − ( α ∗ ) ∂ β b i − ( β ) ϑ − β ( β ∗ − β ) − [ nτ β ∗ ] n n n (cid:88) i =[ nτ β ∗ ]+1 T d a − i − ( α ∗ ) ∂ β b i − ( β ) ϑ − β ( β ∗ − β ) + o p (1) p −→ τ β ∗ (1 − τ β ∗ ) (cid:90) R d T d a − ( x, α ∗ ) ∂ β b ( x, β )d µ ( α ∗ ,β ) ( x )( d − d ) , and (cid:32) − [ nτ β ∗ ] n (cid:33) T ϑ β ) nτ β ∗ ] (cid:88) i =1 E β ∗ [ ξ i | G ni − ] − (cid:32) [ nτ β ∗ ] n (cid:33) T ϑ β ) n (cid:88) i =[ nτ ∗ ]+1 E β ∗ [ ξ i | G ni − ]= 1 T ϑ β (cid:32) − [ nτ β ∗ ] n (cid:33) T [ nτ β ∗ ] (cid:88) i =1 E β ∗ [ ξ i | G ni − ] − (cid:32) [ nτ β ∗ ] n (cid:33) T n (cid:88) i =[ nτ ∗ ]+1 E β ∗ [ ξ i | G ni − ] p −→ . Therefore, from Lemma 9 of Genon-Catalot and Jacod (1993), we see1
T ϑ β [ nτ β ∗ ] (cid:88) i =1 ξ i − [ nτ β ∗ ] n T ϑ β n (cid:88) i =1 ξ i p −→ τ β ∗ (1 − τ β ∗ ) (cid:90) R d T d a − ( x, α ∗ ) ∂ β b ( x, β )d µ β ( x )( d − d ) . Hence, from (7.55) and this, we obtain1
T ϑ β [ nτ β ∗ ] (cid:88) i =1 ˆ ξ i − [ nτ β ∗ ] n T ϑ β n (cid:88) i =1 ˆ ξ i p −→ τ β ∗ (1 − τ β ∗ ) (cid:90) R d T d a − ( x, α ∗ ) ∂ β b ( x, β )d µ β ( x )( d − d ) (cid:54) = 0 , which implies T β ,n −→ ∞ from (7.54) and T ϑ β −→ ∞ , that is, P ( T β ,n > w ( (cid:15) )) −→ (cid:3) Proof of Proposition 3.
We see T β ,n = 1 √ T max ≤ k ≤ n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) I − / n (cid:32) k (cid:88) i =1 ˆ ζ i − kn n (cid:88) i =1 ˆ ζ i (cid:33)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≥ √ T (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) I − / n [ nτ β ∗ ] (cid:88) i =1 ˆ ζ i − [ nτ β ∗ ] n n (cid:88) i =1 ˆ ζ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:113) T ϑ β (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) I − / n T ϑ β [ nτ β ∗ ] (cid:88) i =1 ˆ ζ i − [ nτ β ∗ ] n T ϑ β n (cid:88) i =1 ˆ ζ i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . (7.56)Now we can expressˆ ζ i = ∂ β b i − ( ˆ β ) T A − i − (ˆ α )(∆ X i − hb i − ( ˆ β ))= ∂ β b i − ( ˆ β ) T A − i − ( α ∗ )(∆ X i − hb i − ( ˆ β ))+ (cid:90) ∂ α (cid:16) ∂ β b i − ( ˆ β ) T A − i − ( α )(∆ X i − hb i − ( ˆ β )) (cid:17)(cid:12)(cid:12)(cid:12) α = α ∗ + u (ˆ α − α ∗ ) d u (ˆ α − α ∗ )= ∂ β b i − ( ˆ β ) T A − i − ( α ∗ )(∆ X i − hb i − ( ¯ β ∗ )) − h∂ β b i − ( ˆ β ) T A − i − ( α ∗ )( b i − ( ˆ β ) − b i − ( ¯ β ∗ ))+ (cid:90) ∂ α (cid:16) ∂ β b i − ( ˆ β ) T A − i − ( α )(∆ X i − hb i − ( ˆ β )) (cid:17)(cid:12)(cid:12)(cid:12) α = α ∗ + u (ˆ α − α ∗ ) d u (ˆ α − α ∗ )= ∂ β b i − ( β ∗ ) T A − i − ( α ∗ )(∆ X i − hb i − ( ¯ β ∗ )) + M − (cid:88) j =1 ∂ jβ (cid:16) ∂ β b i − ( β ) T A − i − ( α ∗ )(∆ X i − hb i − ( ¯ β ∗ )) (cid:17)(cid:12)(cid:12)(cid:12) β = ¯ β ∗ ⊗ ( ˆ β − ¯ β ∗ ) ⊗ j + (cid:90) ∂ Mβ (cid:16) ∂ β b i − ( β ) T A − i − ( α ∗ )(∆ X i − hb i − ( ¯ β ∗ )) (cid:17)(cid:12)(cid:12)(cid:12) β = ¯ β ∗ + u ( ˆ β − ¯ β ∗ ) ⊗ ( ˆ β − ¯ β ∗ ) ⊗ M − h (cid:90) ∂ β b i − ( ˆ β ) T A − i − ( α ∗ ) ∂ β b i − ( ¯ β ∗ + u ( ˆ β − ¯ β ∗ ))d u ( ˆ β − ¯ β ∗ )+ (cid:90) ∂ α (cid:16) ∂ β b i − ( ˆ β ) T A − i − ( α )(∆ X i − hb i − ( ˆ β )) (cid:17)(cid:12)(cid:12)(cid:12) α = α ∗ + u (ˆ α − α ∗ ) d u (ˆ α − α ∗ )=: ζ i + M − (cid:88) j =1 ∂ jβ (cid:16) ∂ β b i − ( β ) T A − i − ( α ∗ )(∆ X i − hb i − ( ¯ β ∗ )) (cid:17)(cid:12)(cid:12)(cid:12) β = ¯ β ∗ ⊗ ( ˆ β − ¯ β ∗ ) ⊗ j + (cid:90) ∂ Mβ (cid:16) ∂ β b i − ( β ) T A − i − ( α ∗ )(∆ X i − hb i − ( ¯ β ∗ )) (cid:17)(cid:12)(cid:12)(cid:12) β = ¯ β ∗ + u ( ˆ β − ¯ β ∗ ) ⊗ ( ˆ β − ¯ β ∗ ) ⊗ M − h (cid:90) ∂ β b i − ( ˆ β ) T A − i − ( α ∗ ) ∂ β b i − ( ¯ β ∗ + u ( ˆ β − ¯ β ∗ ))d u ( ˆ β − ¯ β ∗ )+ (cid:90) ∂ α (cid:16) ∂ β b i − ( ˆ β ) T A − i − ( α )(∆ X i − hb i − ( ˆ β )) (cid:17)(cid:12)(cid:12)(cid:12) α = α ∗ + u (ˆ α − α ∗ ) d u (ˆ α − α ∗ ) . Thus, we have1
T ϑ β [ nτ β ∗ ] (cid:88) i =1 ˆ ζ i − [ nτ β ∗ ] n T ϑ β n (cid:88) i =1 ˆ ζ i = 1 T ϑ β [ nτ β ∗ ] (cid:88) i =1 ζ i − [ nτ β ∗ ] n T ϑ β n (cid:88) i =1 ζ i + o p (1)= (cid:32) − [ nτ β ∗ ] n (cid:33) T ϑ β [ nτ β ∗ ] (cid:88) i =1 ζ i − [ nτ β ∗ ] n T ϑ β n (cid:88) i =[ nτ β ∗ ]+1 ζ i + o p (1) , (7.57) (cid:32) − [ nτ β ∗ ] n (cid:33) T ϑ β [ nτ β ∗ ] (cid:88) i =1 E β ∗ [ ζ i | G ni − ] − [ nτ β ∗ ] n T ϑ β n (cid:88) i =[ nτ β ∗ ]+1 E β ∗ [ ζ i | G ni − ]= (cid:32) − [ nτ β ∗ ] n (cid:33) T ϑ β [ nτ β ∗ ] (cid:88) i =1 h∂ β b i − ( ¯ β ∗ ) T A − i − ( α ∗ )( b i − ( β ∗ ) − b i − ( ¯ β ∗ )) − [ nτ β ∗ ] n T ϑ β n (cid:88) i =[ nτ β ∗ ]+1 h∂ β b i − ( ¯ β ∗ ) T A − i − ( α ∗ )( b i − ( β ∗ ) − b i − ( ¯ β ∗ )) + O p ( h/ϑ β )= (cid:32) − [ nτ β ∗ ] n (cid:33) nϑ β [ nτ β ∗ ] (cid:88) i =1 ∂ β b i − ( ¯ β ∗ ) T A − i − ( α ∗ )( b i − ( β ) − b i − ( ¯ β ∗ )) − [ nτ β ∗ ] n nϑ β n (cid:88) i =[ nτ β ∗ ]+1 ∂ β b i − ( ¯ β ∗ ) T A − i − ( α ∗ )( b i − ( β ) − b i − ( ¯ β ∗ ))+ (cid:32) − [ nτ β ∗ ] n (cid:33) nϑ β [ nτ β ∗ ] (cid:88) i =1 ∂ β b i − ( ¯ β ∗ ) T A − i − ( α ∗ ) ∂ β b i − ( β )( β ∗ − β ) − [ nτ β ∗ ] n nϑ β n (cid:88) i =[ nτ β ∗ ]+1 ∂ β b i − ( ¯ β ∗ ) T A − i − ( α ∗ ) ∂ β b i − ( β )( β ∗ − β )+ (cid:32) − [ nτ β ∗ ] n (cid:33) nϑ β [ nτ β ∗ ] (cid:88) i =1 ∂ β b i − ( ¯ β ∗ ) T A − i − ( α ∗ ) (cid:90) (1 − u ) ∂ β b i − ( β + u ( β ∗ − β ))d u ⊗ ( β ∗ − β ) ⊗ − [ nτ β ∗ ] n nϑ β n (cid:88) i =[ nτ β ∗ ]+1 ∂ β b i − ( ¯ β ∗ ) T A − i − ( α ∗ ) (cid:90) (1 − u ) ∂ β b i − ( β + u ( β ∗ − β ))d u ⊗ ( β ∗ − β ) ⊗ + o p (1)= − (cid:32) − [ nτ β ∗ ] n (cid:33) nϑ β [ nτ β ∗ ] (cid:88) i =1 ∂ β b i − ( ¯ β ∗ ) T A − i − ( α ∗ ) (cid:18) ∂ β b i − ( β )( ¯ β ∗ − β )+ (cid:90) (1 − u ) ∂ β b i − ( β + u ( ¯ β ∗ − β ))d u ⊗ ( ¯ β ∗ − β ) ⊗ (cid:19) + [ nτ β ∗ ] n nϑ β n (cid:88) i =[ nτ β ∗ ]+1 ∂ β b i − ( ¯ β ∗ ) T A − i − ( α ∗ ) (cid:18) ∂ β b i − ( β )( ¯ β ∗ − β )+ (cid:90) (1 − u ) ∂ β b i − ( β + u ( ¯ β ∗ − β ))d u ⊗ ( ¯ β ∗ − β ) ⊗ (cid:19) + (cid:32) − [ nτ β ∗ ] n (cid:33) n [ nτ β ∗ ] (cid:88) i =1 ∂ β b i − ( ¯ β ∗ ) T A − i − ( α ∗ ) ∂ β b i − ( β ) ϑ − β ( β ∗ − β ) − [ nτ β ∗ ] n n n (cid:88) i =[ nτ β ∗ ]+1 ∂ β b i − ( ¯ β ∗ ) T A − i − ( α ∗ ) ∂ β b i − ( β ) ϑ − β ( β ∗ − β ) + o p (1)= − (cid:32) − [ nτ β ∗ ] n (cid:33) n [ nτ β ∗ ] (cid:88) i =1 ∂ β b i − ( ¯ β ∗ ) T A − i − ( α ∗ ) ∂ β b i − ( β ) ϑ − β ( ¯ β ∗ − β )+ [ nτ β ∗ ] n n n (cid:88) i =[ nτ β ∗ ]+1 ∂ β b i − ( ¯ β ∗ ) T A − i − ( α ∗ ) ∂ β b i − ( β ) ϑ − β ( ¯ β ∗ − β )+ (cid:32) − [ nτ β ∗ ] n (cid:33) n [ nτ β ∗ ] (cid:88) i =1 ∂ β b i − ( ¯ β ∗ ) T A − i − ( α ∗ ) ∂ β b i − ( β ) ϑ − β ( β ∗ − β ) − [ nτ β ∗ ] n n n (cid:88) i =[ nτ β ∗ ]+1 ∂ β b i − ( ¯ β ∗ ) T A − i − ( α ∗ ) ∂ β b i − ( β ) ϑ − β ( β ∗ − β ) + o p (1) p −→ τ β ∗ (1 − τ β ∗ ) (cid:90) R d ∂ β b ( x, β ) T A − ( x, α ∗ ) ∂ β b ( x, β )d µ β ( x )( d − d ) , and (cid:32) − [ nτ β ∗ ] n (cid:33) T ϑ β ) nτ β ∗ ] (cid:88) i =1 E β ∗ [ ζ i | G ni − ] − (cid:32) [ nτ β ∗ ] n (cid:33) T ϑ β ) n (cid:88) i =[ nτ ∗ ]+1 E β ∗ [ ζ i | G ni − ]= 1 T ϑ β (cid:32) − [ nτ β ∗ ] n (cid:33) T [ nτ β ∗ ] (cid:88) i =1 E β ∗ [ ζ i | G ni − ] − (cid:32) [ nτ β ∗ ] n (cid:33) T n (cid:88) i =[ nτ ∗ ]+1 E β ∗ [ ζ i | G ni − ] p −→ . Therefore, from Lemma 9 of Genon-Catalot and Jacod (1993), we see1
T ϑ β [ nτ β ∗ ] (cid:88) i =1 ζ i − [ nτ β ∗ ] n T ϑ β n (cid:88) i =1 ζ i p −→ τ β ∗ (1 − τ β ∗ ) (cid:90) R d ∂ β b ( x, β ) T A − ( x, α ∗ ) ∂ β b ( x, β )d µ β ( x )( d − d )Hence, from (7.57) and this, we obtain1 T ϑ β [ nτ β ∗ ] (cid:88) i =1 ˆ ζ i − [ nτ β ∗ ] n T ϑ β n (cid:88) i =1 ˆ ζ i p −→ τ β ∗ (1 − τ β ∗ ) (cid:90) R d ∂ β b ( x, β ) T A − ( x, α ∗ ) ∂ β b ( x, β )d µ β ( x )( d − d ) (cid:54) = 0 , which implies T β ,n −→ ∞ from (7.56) and T ϑ β −→ ∞ , that is, P ( T β ,n > w q ( (cid:15) )) −→ (cid:3) acknowledgements This work was partially supported by JST CREST Grant Number JPMJCR14D7 and JSPS KAKENHIGrant Number JP17H01100.
References
Cs¨org¨o, M., Horv´ath, L. (1997). Limit Theorems in Change-Point Analysis, Wiley, New York.De Gregorio, A., Iacus, S. M. (2008). Least squares volatility change point estimation for partially observed diffusion processes.Communications in Statistics - Theory and Methods, , 2342-2357.Genon-Catalot, V., Jacod, J. (1993). On the estimation of the diffusion coefficient for multidimensional diffusion processes.Annales de l’Institut Henri Poincar´e Probabilit´es et Statistiques, , 119-151.Hall, P., Heyde, C. C. (1980). Martingale limit theory and its application. Academic Press, New York.Iacus, S. M., Yoshida, N. (2012). Estimation for the change point of volatility in a stochastic differential equation. StochasticProcesses and their Applications, , 1068-1092.Kessler, M. (1995). Estimation des param`etres d’une diffusion par des contrastes corrig´es. Comptes rendus de l’Acad´emie dessciences. Paris Serie I , 359-362.Kessler, M. (1997). Estimation of an Ergodic Diffusion from Discrete Observations. Scandinavian Journal of Statistics, ,211-229.Lee, S. (2011). Change point test for dispersion parameter based on discretely observed sample from SDE models. Bulletin ofthe Korean Mathematical Society, , 839-845.McLeish, D., L. (1974). Dependent central limit theorems and invariance principles. The Annals of Probability, , 620-628.Negri, I., Nishiyama, Y. (2017). Z-process method for change point problems with applications to discretely observed diffusionprocesses. Statistical Methods and Applications, , 106832.Song, J., Lee, S. (2009). Test for parameter change in discretely observed diffusion processes. Statistical Inference for StochasticProcesses, , 165-183.Tonaki, Y., Kaino, Y., Uchida, M. (2020). Adaptive tests for parameter changes in ergodic diffusion processes from discreteobservations. arXiv:2004.13998.Uchida, M., Yoshida, N. (2011). Estimation for misspecified ergodic diffusion processes from discrete observations. EuropeanSeries in Applied and Industrial Mathematics: Probability and Statistics, Volume 15, 270-290.Uchida, M., Yoshida, N. (2012). Adaptive estimation of an ergodic diffusion process based on sampled data. Stochastic Processesand their Applications,122