Divergence of an integral of a process with small ball estimate
aa r X i v : . [ m a t h . P R ] F e b Divergence of an integral of a process with small ballestimate ⋆ Yuliya Mishura a , Nakahiro Yoshida b,c a Taras Shevchenko National University of Kyiv b Graduate School of Mathematical Sciences, University of Tokyo c Japan Science and Technology Agency CREST
Abstract
The paper contains sufficient conditions on the function f and the stochas-tic process X that supply the rate of divergence of the integral functional R T f ( X t ) dt at the rate T − ǫ as T → ∞ for every ǫ >
0. These conditions in-clude so called small ball estimates which are discussed in detail. Statisticalapplications are provided.
Keywords: integral functional, rate of divergence, small ball estimate,statistical applications
1. Introduction
The problem of convergence or divergence of perpetual integral function-als Z ∞ g ( X ( t )) d t for several classes of stochastic processes and several classes of functions g appears when studying a variety of issues. Let X = { X ( t ) , t ≥ } bea one-dimensional stochastic process with continuous trajectories, and let ⋆ This work was in part supported by Japan Science and Technology Agency CRESTJPMJCR14D7; Japan Society for the Promotion of Science Grants-in-Aid for ScientificResearch No. 17H01702 (Scientific Research); and by a Cooperative Research Program ofthe Institute of Statistical Mathematics.
Email addresses: [email protected] (Yuliya Mishura), [email protected] (Nakahiro Yoshida)
Preprint submitted to Elsevier February 3, 2021 : R → R be a continuous function. Then for any T > Z T g (cid:0) X ( t ) (cid:1) d t (1.1)is defined. However, its properties and asymptotic behavior as T → ∞ de-pend crucially on the properties of the process X and of function g . Theasymptotic behavior of the integral functional R T g ( X ( t )) d t is very differenteven for one-dimensional Markov processes and depends on their transientor recurrent properties. On the one hand, conditions of existence of theperpetual integral functionals and in the case of non-existence, the rate ofdivergence, were studied, in the stochastic framework, for various classesof one- and multidimensional semimartingale and only partially for non-semimartingale stochastic processes, e.g, in [2, 4, 5, 6, 7, 10, 11, 17]. In thepapers where the rate of the divergence was studied, corresponding normal-izing factors were suitable for the central or other functional limit theorems.On the other hand, this question arises in the parametric statistical esti-mation because such integral functionals appear as the denominators of theresidual terms, see, e.g., [8] for fractional models and [9] both for Wiener dif-fusions and fractional diffusions. In such case we need the divergence of thisintegral with some fixed rate and with probability 1, in order to get stronglyconsistent estimators. In this connection, the aim of the present paper isto investigate the rate of convergence of the integral R T f ( X t ) dt to infinityas T → ∞ depending on the properties of the measurable function f andstochastic process X . It is well-known that in the case when X is stationary,ergodic and E f ( X ) is finite, then T − R T f ( X t ) dt → E f ( X ) as T → ∞ ,and then the rate of divergence is evident, T − ǫ R T f ( X t ) dt → ∞ for any ǫ >
0. If process X is not ergodic, the situation is more involved and theconditions on f and X should be much more complicated. In our approach,these conditions include so called small ball estimates. Since these condi-tions are interesting both themselves and from the point of view of variousapplications, see e.g. [13], we consider them even in those examples wherethe processes are ergodic, see examples 2.5 and 2.6.The paper is organized as follows. Section 2 contains the basic conditionsfor the function f and for the process X . Conditions for X include smallball estimates which are presented in two versions, more mild and then morestrong, with various examples. In Section 3 the main divergence theorem isproved, and then two modifications and several examples are provided. Sec-2ion 4 presents two statistical applications. Section contains some auxiliaryresults.
2. Main conditions. Discussion of small ball estimates
Let X = { X t , t ≥ } be a real-valued stochastic process, and f = f ( x ) : R → R be a measurable function. Let the following assumption hold:(A) For any T > I T = Z T f ( X t ) dt is correctly defined. Our goal is to establish sufficient conditions on func-tion f and process X that supply the divergence to infinity: T − ǫ I T → ∞ a.s. (2.1)as T → ∞ for every ǫ > f andstochastic process X , and respective assumptions will be non-trivial, let usconsider them separately, with comments and examples. f Concerning function f , introduce the following notations: denote the sets H + ( x, η ) = [ x, x + η ) and H − ( x, η ) = ( x − η, x ]. Basic assumptions on function f will be as follows.(A1) (i) There exist positive constants K and η ∗ such that K ( η ) := inf x ∈ R min H ∈{ H + ,H − } sup y ∈ H ( x,η ) (cid:12)(cid:12) f ( y ) (cid:12)(cid:12) ≥ η K for every η ∈ (0 , η ∗ ).(ii) Function f is from class C ( R ) and for some constant C we havethat | f ′ ( x ) | ≤ C (1 + | x | ) C ( x ∈ R ) . Let us consider the equivalent form of assumption ( A , ( i ), sufficientcondition for it and the simplest examples and a counterexample.3 emma 2.1. (1) Assumption (A1), (i) is equivalent to any of the followingconditions:(A3) There exist positive constants K, C and η ∗ such that K ( η ) := inf x ∈ R min H ∈{ H + ,H − } sup y ∈ H ( x,η ) (cid:12)(cid:12) f ( y ) (cid:12)(cid:12) ≥ Cη K for every η ∈ (0 , η ∗ ) .(A4) There exist positive constants K and η ∗ such that K ( η ) := inf x ∈ R sup y ∈ ( x,x + η ) (cid:12)(cid:12) f ( y ) (cid:12)(cid:12) ≥ η K for every η ∈ (0 , η ∗ ) .(A5) There exist positive constants K, C and η ∗ such that K ( η ) := inf x ∈ R sup y ∈ ( x,x + η ) (cid:12)(cid:12) f ( y ) (cid:12)(cid:12) ≥ Cη K for every η ∈ (0 , η ∗ ) .(2) Let there exist such positive constants η , δ and d ∈ N such that f ∈ C ( d ) ( R ) and inf x ∈ R max ≤ i ≤ d inf y ∈ [ x,x + η ] (cid:12)(cid:12) f ( i ) ( y ) (cid:12)(cid:12) ≥ δ. (2.2) Then assumption ( A , ( i ) holds.Proof. (1) Let us prove equivalence of conditions ( A , ( i ) and ( A A , ( i ) holds then ( A
3) holds with C = 1. Inversely, if ( A C ≥ A , ( i ) obviously holds, and if ( A
3) holds with
C <
1, we can take K ′ = K + 1 and η ′ ∗ = η ∗ ∧ C . Equivalence of ( A A
5) can be established similarly. Now, let us prove equivalence of( A , ( i ) and ( A A , ( i ) hold. Since K ( η ) ≥ K ( η/ K ( η ) ≥ − K η K , whence ( A
5) consequently ( A
4) holds. Inversely,let ( A
4) hold. Then both values inf x ∈ R sup y ∈ H − ( x,η ) (cid:12)(cid:12) f ( y ) (cid:12)(cid:12) ≥ K ( η ) ≥ η K and inf x ∈ R sup y ∈ H + ( x,η ) (cid:12)(cid:12) f ( y ) (cid:12)(cid:12) ≥ K ( η ) ≥ η K , whence K ( η ) ≥ η K .(2) Let assumption (2.2) hold. Let us fix x ∈ R . Without loss of generality,4ssume that inf y ∈ ( x,x + η ) (cid:12)(cid:12) f ( d ) ( y ) (cid:12)(cid:12) ≥ δ. If, additionally, inf y ∈ ( x,x + η ) (cid:12)(cid:12) f ( d − ( y ) (cid:12)(cid:12) ≥ δη , then we proceed with f ( d − . If inf y ∈ ( x,x + η ) (cid:12)(cid:12) f ( d − ( y ) (cid:12)(cid:12) < δη , then we check in which of fourintervals [ x, x + η / , [ x + η / , x + η / , [ x + η / , x + 3 η /
4] or [ x +3 η / , x + η ] there exists a point y satisfying inequality (cid:12)(cid:12) f ( d − ( y ) (cid:12)(cid:12) < δη . Let, for example, y ∈ [ x + η / , x + η / . Then for any z ∈ [ x + 3 η / , x + η ] we have that | f ( d − ( z ) − f ( d − ( y ) | ≥ δη , there-fore for any z ∈ [ x + 3 η / , x + η ] we have that | f ( d − ( z ) | ≥ δη . Then we continue the same way with | f ( d − | , and in the worst case,the smallest value that we can obtain, is: | f ( z ) | ≥ δη d d for z in someinterval of the diameter η d d . However, even this worst case means thatwe can put in assumption η ∗ = η d , K = d and C = δ d in assumption( A A , ( i ), as it was already established. So,the proof follows. Example 2.2.
Consider the classes of functions satisfying assumption (2.2) .Obviously, any polynomial function P m ( x ) of m th power satisfies (2.2) be-cause at least one of its derivatives is a non-zero constant. Also, any linearcombination of the form P ki =1 ( a i sin( α i x ) + b i cos( β i x )) satisfies this assump-tion as well as the rational function P m ( x ) Q n ( x ) with m > n and Q n ( x ) = 0 . Anexample of f that satisfies ( A is f ( x ) = 1 { x =0 } x sin (cid:18) x (cid:19) . A periodic version f ( x ) = 1 { x π Z } (cid:0) sin x (cid:1) × sin (cid:18) x (cid:19) is also an example that satisfies ( A and has infinitely many clusters of nullpoints in every neighborhood of ∞ . Exponential function e x does not satisfythis assumption around −∞ . .2. Assumptions on process X , with examples The first group of assumptions describes the processes bounded in L ∞− .It is formulated as follows.(A2) (i) (H¨older continuity in L ∞− ) X is continuous a.s. and there exits apositive constant ρ such thatsup s,t ∈ R + : s
Consider the class of processes satisfying assumption ( A , ( iii ) (relaxed small ball estimate). In order to do this, let us combine Theorem 4.4from [12] with assumptions from [13]. More precisely, let X = { X t , t ≥ } be a centered Gaussian process. We assume now that its variance distancesatisfies two-sided power bounds: there exist H ∈ (0 , , and C , C , C > such that for any s, t ≥ , | t − s | ≤ C we have that C | t − s | H ≤ E ( X t − X s ) ≤ C | t − s | H . (2.3) Let us work within this assumption. Note that Theorem 4.4 [12], in a little bitadapted form, states the following: let { Z t , t ∈ [0 , ∆] } be a centered Gaussian rocess. Then for any < a ≤ / and η > P (cid:26) sup ≤ t ≤ ∆ | Z t | ≤ η (cid:27) ≤ exp ( − η a P ≤ i,j ≤ /a ( Eξ i ξ j ) ) , provided that a P ≤ i ≤ /a Eξ i ≥ η , where ξ i = Z ia ∆ − Z ( i − a ∆ . Now let usfix s ≥ , ∆ > , and put Z t = X t + s − X s , ≤ t ≤ ∆ . Then ξ i = Z ia ∆ − Z ( i − a ∆ = X ia ∆ − X ( i − a ∆ , and it follows from assumption (2.3) that C ( a ∆) H ≤ E ξ i ≤ C ( a ∆) H , and so the inequality a P ≤ i ≤ /a E ξ i ≥ η is fulfilled if C ( a ∆) H ≥ η ,or, that is the same, a ≥ √ √ C ! /H η /H ∆ . (2.4) Together with the inequality a ≤ / we get that η ≤ C ∆ H , where C = √ C H √ . Additionally, we assume that the increments of X arepositively correlated, more exactly, for any s i , t i ∈ R + , i = 1 , , s ≤ t ≤ s ≤ t E ( X t − X s ) ( X t − X s ) ≥ . (2.5) Note that positive correlation immediately implies that X ≤ i,j ≤ a ( E ξ i ξ j ) ≤ max ≤ i,j ≤ a E ξ i ξ j E ( X ∆+ s − X s ) ≤ max ≤ i ≤ a E ξ i C ∆ H ≤ C a H ∆ H . (2.6) Now put ∆ ∗ = C , η ∗ = 1 , ∆ ≤ ∆ ∗ , η ≤ C ∆ H , a = C η /H ∆ , where C =7 √ √ C (cid:17) /H . Then η a P ≤ i,j ≤ a ( E ξ i ξ j ) ≥ η C a H ∆ H ≥ η C C H η /H +2 ∆ H − = C ∆ − H η /H − , where C = C C H . It means that assumption ( A , ( iii ) holds with ∆ ∗ = C , η ∗ = 1 , K = 1 , K = C , K = C , γ = H , µ = 2 − H , λ = H − .Evidently, assumptions (2.3) and (2.5) hold for fractional Brownian mo-tion with H > . According to [3] and [13], a subfractional Brownian motionwith H > also satisfies (2.3) and (2.5) . Note, however, that assumption ( A , ( ii ) fails for these processes. The next example supplies us with four classes of the processes satisfyingall assumptions ( A , ( i ) − ( iii ). Example 2.4. (Periodic Brownian bridge) Consider a process that in somesense is a periodic Brownian bridge. Namely, let X ( k ) = { X ( k ) t , t ∈ [ k, k +1) } be a sequence of independent Brownian bridges, constructed between thepoints ( k, and ( k + 1 , , k ≥ . They satisfy the relation of a form X ( k ) t = ( k + 1 − t ) Z tk dW ( k ) u k + 1 − u , t ∈ [ k, k + 1) , where W ( k ) , k ≥ is a sequence of independent Wiener processes, and let X t = X ( k ) t , t ∈ [ k, k + 1) } . Evidently, we constructed a Gaussian process, andsimple calculations show that its characteristics equal E X t = 0 , E (cid:16) X ( k ) t (cid:17) = ( t − k )( k +1 − t ) , E (cid:16) X ( k ) t − X ( k ) s (cid:17) = ( t − s )(1+ s − t ) . The middle equality means that assumption ( A , ( ii ) holds, while last equalitymeans that for s, t ∈ [ k, k + 1) and ≤ t − s ≤ / we have that t − s ≤ E ( X t − X s ) ≤ t − s. onsider now t and s from neighbor intervals, and let s ∈ [ k − , k ) , t ∈ [ k, k + 1) and t − s < / . Then, on the one hand, we have the followingrelations: E ( X t − X s ) = E (cid:16) X ( k ) t (cid:17) + E (cid:0) X ( k − s (cid:1) = t − s − ( t − k ) − ( s − k ) ≤ t − s. On the other hand, for s ≤ k ≤ t we have that ( t − k ) + ( s − k ) ≤ ( t − s ) ,and for t − s < / we have the inequality E ( X t − X s ) ≥ t − s − ( t − s ) ≥ t − s . In particular, it means that E ξ i ≥ a ∆2 ≥ η consequently the inequality a P ≤ i ≤ /a E ξ i ≥ η holds provided that a < / and η ≤ ∆ / √ . All the relations above supply assumption ( A , ( i ) andrelations (2.3) with ρ = H = 1 / . Note that the increments of X are notpositively, but negatively correlated. Consider only interval [0 , , other casescan be treated similarly. On this interval, it is easy to see that for any ≤ s ≤ t ≤ u ≤ v ≤ E ( X t − X s ) ( X v − X u ) = − ( v − u )( t − s ) < . In view on negative correlation of increments, we can not apply up-per bound (2.6) . However, we can calculate and evaluate the sum S := P ≤ i,j ≤ a ( E ξ i ξ j ) explicitly: S = X ≤ i ≤ a ( E ( ξ i ) ) + X ≤ i,j ≤ a ,i = j ( E ξ i ξ j ) = (cid:18) a − (cid:19) ( a ∆(1 − a ∆)) + (cid:18) a − (cid:19) − (cid:18) a − (cid:19)! a ∆ ≤ a ∆ + a ∆ . (2.7) Furthermore, a < / , ∆ < , therefore, S < a ∆ . Therefore, taking intoaccount (2.4) with H = 1 / and considering a = C η ∆ with η ≤ C ∆ / , we et η a P ≤ i,j ≤ a ( E ξ i ξ j ) = η a S ≥ η a ∆ ≥ ∆32 C η . (2.8) It means that assumption ( A , ( iii ) holds with ∆ ∗ = 1 , K = 1 , K = C , K = C , γ = 1 / , µ = 1 , λ = 2 . Example 2.5. (Stationary Ornstein–Uhlenbeck process) Consider even moresimple and natural example. Having in mind calculations from Example 2.4,we can omit some technical details. So, introduce a stationary Ornstein–Uhlenbeck process of the form X t = Z t −∞ e θ ( s − t ) dW s , where W is a two-sided Wiener process, θ > . For the technical simplic-ity, we put θ = 1 . Then E X t = , and this process is Gaussian, thereforecondition ( A , ( ii ) holds, X t − X s = (cid:0) e − t − e − s (cid:1) Z s −∞ e z dW z + e − t Z ts e z dW z , for any s < t , whence E ( X t − X s ) = 1 − e s − t . Evidently,on the one hand, − e s − t ≤ t − s . On the other hand, we canstate that − e − x = e − θ ( t − s ) ≥ e − ( t − s ) if t − s < (here θ ∈ ( − x, ).Therefore two-sided inequality (2.3) holds with H = 1 / , and assumption ( A , ( i ) holds. In addition, Moreover, for any s ≤ t ≤ u ≤ v we have that E ( X t − X s ) ( X v − X u ) = 12 (cid:0) e t − e s (cid:1) (cid:0) e − v − e − u (cid:1) < . So, the increments are negatively correlated. Let us evaluate the sum S from , taking into account that if we choose ∆ ∗ = 2 and a = 1 / , then a ∆ < : S = X ≤ i ≤ a ( E ( ξ i ) ) + X ≤ i,j ≤ a ,i = j ( E ξ i ξ j ) = (cid:18) a − (cid:19) (cid:0) − e − a ∆ (cid:1) + 12 (cid:0) − e − a ∆ (cid:1) (cid:0) − e a ∆ (cid:1) X ≤ i Example 2.6. (stationary fractional Ornstein-Uhlenbeck process) Let H ∈ (1 / , , and let B H = { B Ht , t ∈ R } be a two-sided fractional Brownian mo-tion with Hurst index H , that is, a centered Gaussian process with covariancefunction E B Ht B Hs = 12 ( | t | H + | s | H − | t − s | H ) . Let −∞ ≤ a < b ≤ + ∞ , and let a measurable function h : [ a, b ] → R satisfyassumption Z [ a,b ] | h ( u ) || h ( v ) || u − v | H − dudv < ∞ . Then the integral R [ a,b ] h ( z ) dB Hz is correctly defined and is a Gaussian randomvariable with zero mean and variance C H Z [ a,b ] h ( u ) h ( v ) | u − v | H − dudv, C H = H (2 H − . (2.10) In this connection, we can introduce a fractional Ornstein-Uhlenbeck process X t = R t −∞ e θ ( s − t ) dB Hs , θ > , that is a Gaussian stationary process with zeromean. For the technical simplicity, we put θ = 1 . Let’s calculate some ofits quadratic moment characteristics, in order to evaluate the left-hand sideof (2.6) . This evaluation includes several constants whose value is not im-portant, therefore we denote by C various constants whose value can change rom line to line and even inside the same line. First, according to (2.10) , E X t = C H e − t Z ( −∞ ,t ] e u + v | u − v | H − dudv = ( u ′ = − u, v ′ = − v )= C H e − t Z [ − t, ∞ ) e − u ′ − v ′ | u ′ − v ′ | H − du ′ dv ′ = ( u ′ + t = z, v ′ + t = w ) = C H Z R e − z − w | z − w | H − dzdw =: C H I . (2.11) Taking into account Gaussian property of X , we conclude that assumption ( A , ( ii ) holds. Further, for any ≤ s ≤ t applying suitable change ofvariables X t − X s = ( e − t − e − s ) Z s −∞ e z dB Hz + e − t Z ts e z dB Hz , whence E ( X t − X s ) = C H ( e − t − e − s ) Z ( −∞ ,s ] e u + v | u − v | H − dudv +2 C H e − t ( e − t − e − s ) Z s −∞ Z ts e u + v | u − v | H − dudv + C H e − t Z [ s,t ] e u + v | u − v | H − dudv = C H (cid:16) (cid:0) e s − t − (cid:1) I + 2 e s − t (cid:0) e s − t − (cid:1) Z ∞ Z t − s e − u + v | u − v | H − dudv + e s − t Z t − s Z t − s e u + v | u − v | H − dudv (cid:17) . (2.12) Now, on the one hand, taking into account the above relations, the fact that ≤ − e − x ≤ x for x > and evident relation e s − t − < , we can get thatfor any ≤ s ≤ t ≤ E ( X t − X s ) ≤ C H (cid:18) ( t − s ) I + Z [0 ,t − s ] | z − w | H − dzdw (cid:19) ≤ C (cid:0) ( t − s ) I + ( t − s ) H (cid:1) . (2.13) On the other hand, H < , therefore ( e s − t − ) ( t − s ) H → as t → s . Due to Lemma .3, e s − t ( e s − t − R ∞ R t − s e − u + v | u − v | H − dudv ( t − s ) H → as t → s . Finally, e s − t R t − s R t − s e u + v | u − v | H − dudv ( t − s ) H → H (2 H − as t → s . Then it follows from the limit relations above and (2.12) that E ( X t − X s ) ( t − s ) H → as t → s , therefore, there exists d > such that E ( X t − X s ) ≥ 12 ( t − s ) H (2.14) for t − s < d . It follows from (2.13) and (2.14) that on some interval we havetwo sided inequality (2.3) . In particular, it means that assumption ( A , ( i ) holds with ρ = H . Further, as always, we are interested in values s = ia ∆ , t = ( i + 1) a ∆ , ≤ i ≤ a − . In this case we get the following relations ≤ E (cid:0) X ( i +1) a ∆ − X ia ∆ (cid:1) ≤ C (cid:0) a ∆ I + ( a ∆) H (cid:1) . Since H < , for a ∆ < we conclude that ( E ( ξ i )) ≤ C ( I + 1) a H ∆ H ,and the 1st term of the sum S (see (2.7) ) can be bounded as X ≤ i ≤ a ( E ( ξ i )) ≤ C ( I + 1) a H − ∆ H . (2.15) On the other hand, the inequality a X ≤ i ≤ /a E ξ i ≥ η is supplied by the inequality a ≥ (8 η ) /H ∆ . Taking into account that we need toconsider a < / , we can put ∆ ∗ = 2 d and η ≤ ∆ H H . ow, for any ≤ u ≤ v ≤ s ≤ t E ( X t − X s )( X v − X u )= C H (cid:16) ( e − t − e − s )( e − v − e − u ) Z s −∞ Z u −∞ e z + w | z − w | H − dzdw +( e − t − e − s ) e − v Z s −∞ Z vu e z + w | z − w | H − dzdw + e − t ( e − v − e − u ) Z ts Z u −∞ e z + w | z − w | H − dzdw + e − t − v Z ts Z vu e z + w | z − w | H − dzdw (cid:17) = C H (cid:16) ( e s − t − e u − v − Z R e − z − w | z − w + s − u | H − dzdw +( e s − t − Z ∞ Z v − u e − z − w | z − w + s − v | H − dzdw +( e u − v − Z t − s Z ∞ e − z − w | z − w + t − u | H − dzdw + Z t − s Z v − u e − z − w | z − w + t − v | H − dzdw (cid:17) . Taking into account that we are interested in the values u = ia ∆ , v = ( i + 1) a ∆ , s = ja ∆ , t = ( j + 1) a ∆ for some ≤ i < j ≤ a , we get for ξ k = X ( k +1) a ∆ − X ka ∆ , k = i, j that E ξ i ξ j = I (1) ij + 2 I (2) ij + I (4) ij , where I (1) ij = C H ( e − a ∆ − Z R e − z − w | z − w + ( j − i ) a ∆ | H − dzdw,I (2) ij = C H ( e − a ∆ − Z ∞ Z a ∆0 e − z − w | z − w + ( j − i ) a ∆ | H − dzdw,I (3) ij = C H Z a ∆0 Z a ∆0 e − z − w | z − w + ( j − i + 1) a ∆ | H − dzdw. ccording to Lemma 5.2 from Appendix 5, integral Z R e − z − w | z − w + ( j − i ) a ∆ | H − dzdw is bounded by some constant. Therefore, I (1) ij ≤ C ( a ∆) . Furthermore, ac-cording to the L’Hˆospital’s rule lim x → x − Z ∞ Z x e − z − w | z − w + ( j − i ) x | H − dzdw = lim x → Z ∞ e − x − w | x − w + ( j − i ) x | H − dzdw = Z ∞ e − w w H − dw. Therefore, I (2) ij ≤ C ( a ∆) . Finally, and under assumption that a ∆ → I (3) ij = C H Z a ∆0 Z a ∆0 e − z − w | z − w + ( j − i ) a ∆ | H − dzdw ∼ C H Z a ∆0 Z a ∆0 | z − w + ( j − i ) a ∆ | H − dzdw = C H ( a ∆) H Z Z | z − w + ( j − i ) | H − dzdw ∼ C ( a ∆) H , and, according to Lemma 5.1, R R | z − w + ( j − i ) | H − dzdw is bounded bysome constant not depending on j − . Due to the fact that H < , we canstate the following: there exists ∆ ∗ > such that for ∆ < ∆ ∗ , due to thefact that a < / , E ξ i ξ j > . It means that we are exactly in conditions ofExample 2.3 and can produce the same conclusions. Example 2.7. (Tempered fractional Brownian motion) There are severalapproaches how to introduce a tempered fractional Brownian motion. For thedetail see [1, 15, 16]. We shall introduce it as follows. Let θ > , α > .Consider a process Y t = Z t −∞ e − θ ( t − s ) ( t − s ) α dW s , t ≥ . rocess Y is stationary and Gaussian, with the following characteristics: E Y t = 0 , E Y t = Z ∞ e − θz z α dz. As usual, without loss of generality, put θ = 1 . Let us calculate E Y Y t = E Z −∞ e z ( − z ) α dW z Z t −∞ e z − t ( t − z ) α dW z = Z −∞ e z e z − t ( − z ) α ( t − z ) α dz = e − t Z ∞ e − z z α ( t + z ) α dz = t α +1 e − t Z ∞ e − zt z α (1 + z ) α dz → as t → ∞ . So, Y is an ergodic process. We shall not provide the smallball calculations since they are very tedious. Remark 2.8. All examples are about the case where t = 0 . However, sincewe consider asymptotics of the integral, process X can be arbitrary till somefixed t > and satisfy assumption ( A after this moment. Now let us introduce more simple and stronger small ball estimate.(A2), (iv) (stronger small ball estimate) There exist positive constants η ∗ , ∆ ∗ , λ , µ , K and K , such thatsup s ∈ R + P (cid:20) sup t ∈ [ s,s +∆] (cid:12)(cid:12) X t − X s (cid:12)(cid:12) ≤ η (cid:21) ≤ K exp (cid:18) − K η − λ ∆ µ (cid:19) for all η ∈ (0 , η ∗ ) and ∆ ∈ (0 , ∆ ∗ ).As one can see, the difference is that in ( A , ( iii ) η is adapted to ∆, and itmeans that we consider η under the curve η = K ∆ γ , while in ( A , ( iv ) weconsider a whole rectangle η ∈ (0 , η ∗ ) , ∆ ∈ (0 , ∆ ∗ ). All previous examples donot work, however, let us consider two other examples. Example 2.9. One of the simplest examples of the processes X satisfyingassumptions ( A , is X t = ξϕ ( t ) , where ξ is a random variable satisfying thefollowing conditions:(j) All moments of ξ are uniformly bounded; jj) There exist positive constants λ , K ≥ and K , such that for any x > P [ | ξ | ≤ x ] ≤ K exp (cid:0) − K x − λ (cid:1) , and function ϕ is periodic with period 2, and equals ϕ ( t ) = t { ≤ t ≤ } + (2 − t )1 { ≤ t ≤ } . In this case process X is continuous since ϕ is continuous, andcondition ( i ) is fulfilled with ρ = 1 because | ϕ ( t ) − ϕ ( s ) | ≤ | t − s | , condition ( ii ) is supplied by ( j ) because ϕ is a bounded function. Furthermore, sup t ∈ [ s,s +∆] (cid:12)(cid:12) X t − X s (cid:12)(cid:12) ≥ | ξ | ∆2 since sup t ∈ [ s,s +∆] (cid:12)(cid:12) ϕ ( t ) − ϕ ( s ) (cid:12)(cid:12) ≥ ∆2 , for any < ∆ ≤ . Hence sup s ∈ R + P (cid:20) sup t ∈ [ s,s +∆] (cid:12)(cid:12) X t − X s (cid:12)(cid:12) ≤ η (cid:21) = sup s ∈ R + P (cid:20) | ξ | ∆2 ≤ η (cid:21) = sup s ∈ R + P (cid:20) | ξ | ≤ η ∆ (cid:21) ≤ K exp − K (cid:18) η ∆ (cid:19) − λ ! , and so condition ( iii ) follows from ( jj ) with any η ∗ > , < ∆ ∗ < , λ = µ = λ , K = K , K = K λ . In this case we have a small ball estimatein time, but in some sense, uniformly ball estimate in space. We can modifythis example in the following way: let Ω = [0 , , and consider the same ξ butshift the functions ϕ in a random way, namely, let e ϕ ( t, ω ) = ϕ ( t + ω ) . Then sup t ∈ [ s,s +∆] (cid:12)(cid:12) e ϕ ( t, ω ) − e ϕ ( s, ω ) (cid:12)(cid:12) ≥ ∆2 , for any < ∆ ≤ , and we havethe same estimates as before. 3. Divergence theorems The first result describes the conditions of divergence to infinity for theprocesses bounded in L ∞− . We prove it under relaxed small ball estimate( A , ( iii ), however, this theorem is certainly true if to replace relaxed smallball estimate with ( A , ( iv ). 17 heorem 3.1. Let the function f satisfy assumption ( A , ( i ) − ( ii ) , and theprocess X satisfy assumptions ( A , ( i ) − ( iii ) . Then (2.1) holds, i.e., T − ǫ I T → ∞ a.s. as T → ∞ for every ǫ > .Proof. Let ǫ > 0. Fix positive numbers ǫ ( i ) ( i = 0 , , , , 4) such that λǫ (0) > µǫ (1) , ǫ (0) > γǫ (1) , ǫ (1) < ǫ (2) , ǫ (3) < ǫ (2) ρ ,Kǫ (0) < ǫ (3) − C ǫ (4) (3.1)and that 2 Kǫ (0) − ǫ (1) + ǫ (2) < ǫ. Such numbers ǫ ( i ) ( i = 0 , , , , 4) exist; for example, let δ ↓ ǫ (0) = δ , ǫ (1) = δ , ǫ (2) = δ, ǫ (3) = δ , ǫ (4) = δ . Let η n = n − ǫ (0) ∧ (cid:0) K n − γǫ (1) (cid:1) . The 2nd value, K n − γǫ (1) is necessary inorder to apply assumption ( iii ), the relaxed small ball estimate. However, weshall deal with the 1st value, n − ǫ (0) , therefore, assume that n is sufficientlylarge, such that log n > log (cid:16) K (cid:17) ǫ (0) − γǫ (1) . In this case n − ǫ (0) < K n − γǫ (1) , and η n = n − ǫ (0) . Then c n := 14 min (cid:26) inf x ∈ R sup y ∈ H + ( x,η n ) (cid:12)(cid:12) f ( y ) (cid:12)(cid:12) , inf x ∈ R sup y ∈ H − ( x,η n ) (cid:12)(cid:12) f ( y ) (cid:12)(cid:12)(cid:27) ≥ η Kn for large n . Let ∆ n = n − ǫ (1) , s nj = ( j − n and t nj = j ∆ n for j, n ∈ N . Let I nj = [ s nj , s nj + ∆ n / A nj = (cid:26) sup s,t ∈ I nj (cid:12)(cid:12) X t − X s (cid:12)(cid:12) > η n (cid:27) . ω ∈ A nj , there exist τ ( ω ), σ ( ω ) ∈ I nj such that σ ( ω ) < τ ( ω ) andthat (cid:12)(cid:12) X τ ( ω ) ( ω ) − X σ ( ω ) ( ω ) (cid:12)(cid:12) > η n . Therefore, by the mean-value theorem,if X σ ( ω ) ( ω ) < X τ ( ω ) ( ω ), thenmax t ∈ I nj | f ( X t ( ω )) | ≥ sup t ∈ [ σ ( ω ) ,τ ( ω )] | f ( X t ( ω )) | ≥ sup x ∈ H + ( X σ ( ω ) ,η n ) (cid:12)(cid:12) f ( x ) (cid:12)(cid:12) ≥ c n . Similarly, if X τ ( ω ) ( ω ) < X σ ( ω ) ( ω ), then we consider H − ( X σ ( ω ) , η n ) and con-clude that max t ∈ I nj | f ( X t ( ω )) | ≥ c n . (3.2)Thus, inequality (3.2) is always valid for ω ∈ A nj .Let β = ǫ (3) ǫ (2) and let r > ( ρ − β ) − , equivalently, ρ − r > β . By ( A 2) (i), E (cid:2) | X t − X s | r (cid:3) ≤ B ( r ) | t − s | rρ ( t ∈ [ s, s + 1])where B ( r ) = (cid:18) sup s, t ∈ R + : s 0, where M = Z [0 , Z [0 , | t − t | rρ − rβ − dt dt ; (3.4) M is finite since rρ − rβ − > − 1. Sincesup t ∈ [ s,s + n − ǫ (2) ] | X t − X s || t − s | β ≥ n ǫ (2) β sup t ∈ [ s,s + n − ǫ (2) ] | X t − X s | , by setting h = n ǫ (3) in (3.3), and taking into account the definition of β , we19btain sup s ∈ R + P (cid:20) sup t ∈ [ s,s + n − ǫ (2) ] | X t − X s | ≥ n − ǫ (3) (cid:21) ≤ sup s ∈ R + P (cid:20) sup t ∈ [ s,s + n − ǫ (2) ] | X t − X s || t − s | β ≥ n − ǫ (3)+ ǫ (2) β (cid:21) = sup s ∈ R + P (cid:20) sup t ∈ [ s,s + n − ǫ (2) ] | X t − X s || t − s | β ≥ n ǫ (3) (cid:21) ≤ C ( r ) B ( r ) Mn ǫ (3) r (3.5)for all n ∈ N .Obviously, P " sup t ∈ [ s,s + n − ǫ (2) ] (cid:12)(cid:12) f ( X t ) − f ( X s ) (cid:12)(cid:12) ≥ c n ≤ P (cid:2) | X s | ≥ n ǫ (4) (cid:3) + P (cid:20) | X s | ≤ n ǫ (4) , sup t ∈ [ s,s + n − ǫ (2) ] | X t − X s | ≤ n − ǫ (3) , sup t ∈ [ s,s + n − ǫ (2) ] (cid:12)(cid:12) f ( X t ) − f ( X s ) (cid:12)(cid:12) ≥ c n (cid:21) + P " sup t ∈ [ s,s + n − ǫ (2) ] | X t − X s | ≥ n − ǫ (3) . (3.6)By choosing a sufficiently large r in (3.5), we knowsup s ∈ R + P " sup t ∈ [ s,s + n − ǫ (2) ] | X t − X s | ≥ n − ǫ (3) = O ( n − L ) (3.7)as n → ∞ for every L > 0. Moreover, from ( A , ( ii ) we havesup s ∈ R + P (cid:2) | X s | ≥ n ǫ (4) (cid:3) = O ( n − L ) (3.8)as n → ∞ for every L > 0. By Taylor’s formula applied to f , we obtainsup t ∈ [ s,s + n − ǫ (2) ] (cid:12)(cid:12) f ( X t ) − f ( X s ) (cid:12)(cid:12) ≤ sup t ∈ [ s,s + n − ǫ (2) ] C (cid:0) | X s | + | X t − X s | (cid:1) C (cid:12)(cid:12) X t − X s (cid:12)(cid:12) ≤ C (2 + n ǫ (4) ) C n − ǫ (3) | X s | ≤ n ǫ (4) and sup t ∈ [ s,s + n − ǫ (2) ] | X t − X s | ≤ n − ǫ (3) . We also have C (2 + n ǫ (4) ) C n − ǫ (3) < n − Kǫ (0) / η Kn / ≤ c n for large n . Therefore, under ( A , ( ii ) and ( A , ( i ) − ( ii ),sup s ∈ R + P (cid:20) sup t ∈ [ s,s + n − ǫ (2) ] (cid:12)(cid:12) f ( X t ) − f ( X s ) (cid:12)(cid:12) ≥ c n (cid:21) = O ( n − L ) (3.9)as n → ∞ for every L > τ nj = inf { t ≥ s nj ; (cid:12)(cid:12) f ( X t ) (cid:12)(cid:12) ≥ c n } ∧ ( s nj + ∆ n / , and let B nj = ( sup t ∈ [ τ nj ,τ nj + n − ǫ (2) ] (cid:12)(cid:12) f ( X t ) − f ( X τ nj ) (cid:12)(cid:12) < c n ) . (3.10)Let J n = ⌈ n ǫ (1) ⌉ + 1. Now (3.9) gives the estimate P (cid:20)(cid:18) J n \ j =1 B nj (cid:19) c (cid:21) ≤ J n X j =1 ⌈ n − ǫ (1)+ ǫ (2) ⌉ +1 X i =1 P (cid:20) sup t ∈ [ s nj +( i − n − ǫ (2) ,s nj + in − ǫ (2) ] (cid:12)(cid:12) f ( X t ) − f ( X s nj +( i − n − ǫ (2) ) (cid:12)(cid:12) ≥ c n (cid:21) = O ( n − L ) (3.11)as n → ∞ for every L > A , ( iii ) (recall that η n < K ∆ γn ), we havesup j ≤ J n P " sup s,t ∈ I nj | X t − X s | ≤ η n = O ( n − L )21s n → ∞ for every L > 0. That is, P " \ j ≤ J n A nj ! c = O ( n − L ) (3.12)as n → ∞ for every L > A nj ∩ B nj we have that | f ( X t ) | ≥ c n on the interval of length at least n − ǫ (2) , therefore Z [ s nj ,t nj ] f ( X t ) dt ≥ c n n − ǫ (2) . Therefore, I n ≥ n − Kǫ (0) ⌊ n ǫ (1) ⌋ n − ǫ (2) (3.13)on T J n j =1 (cid:0) A nj ∩ B nj (cid:1) . Thanks to (3.11), (3.12) and (3.13) with the inequality2 Kǫ (0) − ǫ (1) + ǫ (2) < ǫ/ 2, we obtain P h I n < n − ǫ i = O ( n − L )as n → ∞ for every L > 0. In particular, by Borel-Cantelli’s lemma, P (cid:20) lim sup n →∞ (cid:8) I n < n − ǫ (cid:9)(cid:21) = 0 . Therefore, P h lim n →∞ (cid:0) n − (1 − ǫ ) I n (cid:1) = ∞ i = 1 . (3.14)This shows (2.1) for T = n but it is sufficient for proof of the theorem. Theorem 3.2. The convergence (2.1) holds if we exclude condition ( A , ( ii ) of boundedness in L ∞− , and instead add the condition ( A iii ) There exist constants Q > , p > and C > such that | f ( x ) | ≥ Q | x | p for any | x | ≥ C . roof. Analyzing proof of Theorem 3.1, we can see that condition ( A , ( ii )is applied only when we construct the upper bound for the first term on theright-hand side of the inequality (3.6). In this connection, we can consider,instead of the first two terms on the left-hand side of (3.6), one term of theform sup s ∈ R + P (cid:20) sup t ∈ [ s,s + n − ǫ (2) ] | X t − X s | ≥ n − ǫ (3) , | X s | ≤ n ǫ (4) (cid:21) = O ( n − L )(3.15)It means that instead of (3.10) we should consider the events e B nj = (cid:26) sup t ∈ [ τ nj ,τ nj + n − ǫ (2) ] (cid:12)(cid:12) f ( X t ) − f ( X τ nj ) (cid:12)(cid:12) < c n (cid:27) [ (cid:26) | X τ nj | ≥ n ǫ (4) (cid:27) . Then the upper bound (3.11) still holds for e B nj in place of B nj , and moreover,on A nj ∩ e B nj we have, as before that | f ( X t ) | ≥ c n on the interval of length atleast n − ǫ (2) , otherwise, for sufficiently large n | X t | ≥ n ǫ (4) − n − ǫ (3) ≥ / n ǫ (4) , whence | f ( X t ) | ≥ Q p n pǫ (4) , therefore, for sufficiently large n Z [ s nj ,t nj ] f ( X t ) dt ≥ c n n − ǫ (2) ∧ Q p n pǫ (4) − ǫ (1) , and we can conclude as in Theorem 3.1.Note that without condition ( A , ( ii ), the diverging rate obtained herecould be far from optimal. This is confirmed by the following statement. Theorem 3.3. Let the function f ( x ) = | x | p , p > , and let the process X = { X t , t ≥ } be a real-valued stochastic process, satisfying the followingconditions ( i ) X is self-similar with index H ∈ (0 , ; ( ii ) The random variable R | X t | p dt satisfies assumption R | X t | p dt ≥ ξ, here ξ is a non-negative random variable with bounded density (par-ticularly, R | X t | p dt itself has a bounded density). Then for any ǫ ∈ (0 , pH ) we have that lim inf T →∞ T − − ǫ Z T | X t | p dt > a.s. Proof. Let constant C > ξ from assumption ( ii ). Then for any k ∈ N , 0 < ǫ < pH , β > x > X that P k,x := P ( R k β | X t | p dtk β (1+ ǫ ) < x ) = P ( R | X sk β | p dsk βǫ < x ) = P (cid:26) k β ( pH − ǫ ) Z | X s | p ds < x (cid:27) = P (cid:26)Z | X s | p ds < xk β ( pH − ǫ ) (cid:27) ≤ P ( ξ < x p k β ( H − ǫp ) ) ≤ C x p k β ( H − ǫp ) . If we choose β > (cid:16) H − ǫp (cid:17) − , then Σ k ≥ P k,x converges, and it followsfrom Borel-Cantelli and the fact that x > k →∞ R k β | X t | p dtk β (1+ ǫ ) = ∞ a.s.Further, for any T ∈ (cid:2) k β , ( k + 1) β ) (cid:3)R T | X t | p dtT ǫ ≥ R k β | X t | p dtk β (1+ ǫ ) (cid:18) kk + 1 (cid:19) β (1+ ǫ ) ≥ β (1+ ǫ ) R k β | X t | p dtk β (1+ ǫ ) , therefore lim inf T →∞ R T | X t | p dtT ǫ = + ∞ a.s.for any 0 < ǫ < pH . 24 xample 3.4. For example, for any p > T →∞ T − − ǫ Z T | B Ht | p dt > a.s. for any 0 < ǫ < pH , where B H is a fractional Brownian motion with Hurstindex H ∈ (0 , B H is a self-similar process with the index H ofself-similarity, and in this case Z | B Ht | p dt ≥ ξ = |N (0 , σ ) | p , and σ = 12 Z Z (cid:0) s H + u H − | s − u | H (cid:1) duds = 12 H + 2 . Obviously, ξ has a bounded density. However, in this particular case wecan say more and establish the exact rate of convergence. Indeed, consider ǫ = pH . In this case, according to Theorem 3.3 [14].lim inf T →∞ sup ≤ s ≤ T | B Hs | p (log log T ) pH T pH = c > , Therefore, P k,x ≤ σ √ π x p k β ( H − ǫp ) . where c is a positive constant. Therefore,lim inf T →∞ R T | B Hs | p dsT pH ≤ lim inf T →∞ sup ≤ s ≤ t | B Hs | p T pH = 0 a.s.A fortiori, for any ǫ > pH lim inf T →∞ R T | B Hs | p dsT (1+ ǫ ) = 0 a.s.Concluding this section, we will consider a stationary X . Let g t = f ( X t ) .Suppose that { g t } t ∈ R + is uniformly integrable, which is satisfied, for example,25nder condition ( A ii ), and that process X is stationary. Then1 T Z T g t dt → Z a.s. (3.16)as T → ∞ for some nonnegative random variable Z by Birkhoff’s individualergodic theorem; Z is a random variable measurable to the invariant σ -field.Define the event A by A = (cid:8) Z = 0 (cid:9) . (3.17)The family (cid:8) T − R T g t dt (cid:9) T > of random variables is uniformly integrable,and hence lim T →∞ T Z T E [ g t A ] dt = lim T →∞ E (cid:20) T Z T g t dt A (cid:21) = E (cid:20) lim T →∞ T Z T g t dt A (cid:21) = E [ Z A ] = 0 . (3.18)Since ( g s , A ) = d ( g t , A ) for any s, t ∈ R + by stationarity of X , (3.18) implies E (cid:2) g t A (cid:3) = 0 (3.19)for any t ∈ R + , and hence, E (cid:20) Z T g t dt A (cid:21) = 0 (3.20)for any T ∈ R + . On the other hand, ifsup T > Z T g t dt > a.s., (3.21)then (3.20) is valid only when P ( A ) = 0, i.e., Z > R T g t dt diverges at the rate of T a.s. as T → ∞ . In particular, under the conditionsof Theorem 3.1, it holds thatlim T →∞ T Z T g t dt > a.s. 4. Statistical Application Let us consider two statistical applications of the divergence results.Namely, let us consider Ornstein–Uhlenbeck process Y = { Y t , t ≥ } withunknown drift parameter θ > Y t = Y − θ Z t Y s ds + Z t g s dW s , where W = { W t , t ≥ } is a Wiener process, g : R + → R is a measurablefunction such that 0 ≤ c ≤ | g s | ≤ C, s ≥ 0. Since Y satisfies assumption( A 2) and f ( x ) = x satisfies assumption ( A T − ǫ R T Y s ds → ∞ and T − ǫ R T Y s g s ds → ∞ for any ǫ > T → ∞ .Consider the equality Z T Y t dY t = − θ Z T Y s ds + Z T Y s g s dW s , whence R T Y t dY t R T Y s ds = − θ + R T Y s g s dW s R T Y s ds . Furthermore, since R T Y s g s ds → ∞ , we get from the strong law of largenumbers for martingales that R T Y s g s dW s R T Y s g s ds → T → ∞ . However, c ≤ R T Y s g s ds R T Y s ds ≤ C , and it means that R T Y s g s dW s R T Y s ds → T → ∞ . 27e get that − R T Y s dY s R T Y s ds is a strongly consistent estimator of θ .Another example can be introduced as follows. Let the processes X and Y be observable, and satisfy the relation X t = X + θ Z t g ( Y s ) ds + B Ht , without any restriction on θ ∈ R and H ∈ (0 , g = f ,where f satisfies ( A 1) and Y satisfies ( A X T R T g ( Y s ) ds = X R T g ( Y s ) ds + θ + B HT R T g ( Y s ) ds . According to [8], B HT T H + ǫ → T → ∞ for any ǫ > R T g ( Y s ) dsT H + ǫ → ∞ a.s. for H + ǫ ≤ 5. Appendix Here we establish some auxiliary results. Lemma 5.1. Let H ∈ (1 / , , x, y ≥ . Then there exists C > dependingonly on H such that Z x | w − y | H − dw ≤ Cx H − . Proof. We consider only x > 0. Let y = 0. Then R x w H − dw = (2 H − − x H − .Let 0 < y ≤ x . Then Z x | w − y | H − dw = Z y ( y − w ) H − dw + Z xy ( w − y ) H − dw = (2 H − − (cid:0) y H − + ( x − y ) H − (cid:1) ≤ H − − x H − . y ≥ x . Then, since | a α − b α | ≤ | a − b | α for α ∈ (0 , Z x | w − y | H − dw = Z x ( y − w ) H − dw = 2 H − − (cid:2) y H − − ( y − x ) H − (cid:3) ≤ (2 H − − x H − . Lemma is proved. Lemma 5.2. There exists such C > that for any p > I := Z R e − z − w | z − w + p | H − dzdw ≤ C. Proof. Let us provide the following transformations: I = Z ∞ e − z Z z + p e − w ( z + p − w ) H − dwdz + Z ∞ e − z Z ∞ z + p e − w ( w − z − p ) H − dwdz = Z ∞ e − z Z z + p e x − z − p x H − dxdz + Z ∞ p e − w Z w − p e − z ( w − p − z ) H − dzdw = e − p (cid:18)Z p e x x H − dx Z ∞ e − z dz + Z ∞ p e x x H − Z ∞ x − p e − z dzdx (cid:19) + Z ∞ p e − w Z w − p e p − w + x x H − dxdw =: I + I + I . Since lim p →∞ e − p Z p e x x H − dx = lim p →∞ e p p H − e p = 0 , the value I = e − p R p e x x H − dx is bounded. Further, I = e − p Z ∞ p e x x H − Z ∞ x − p e − z dzdx = 12 e − p Z ∞ p e x e − x +2 p x H − dx = 12 e p Z ∞ p e − x x H − dx, and lim p →∞ R ∞ p e − x x H − dxe − p = lim p →∞ e − p p H − e − p = 0 , I = e p Z ∞ p e − w Z w − p e x x H − dxdw ≤ (2 H − − Z ∞ p e − w ( w − p ) H − dw = ( w − p = x ) == (2 H − − e − p Z ∞ e − x x H − dx → , p → ∞ . Therefore, this values is bounded, too. Lemma is proved. Lemma 5.3. For any H ∈ (1 / , we have the limit relation lim x → x − Z ∞ Z x e − u + v | u − v | H − dudv = Γ(2 H − . Proof.