Statistical inference for exponential functionals of Lévy processes
SStatistical inference for exponentialfunctionals of L´evy processes ∗ Denis Belomestny
University of Duisburg-EssenThea-Leymann-Str. 9, 45127 Essen, Germanye-mail: [email protected]
Vladimir Panov
National research university Higher School of EconomicsShabolovka, 26, Moscow, 119049 Russia.e-mail: [email protected]
Abstract:
In this paper, we consider the exponential functional A ∞ = (cid:82) ∞ e − ξ s ds of a L´evy process ξ s and aim to estimate the characteristics of ξ s from the distribution of A ∞ . We present a new approach, which allows tostatistically infer on the L´evy triplet of ξ t , and study the theoretical prop-erties of the proposed estimators. The suggested algorithms are illustratedwith numerical simulations. Keywords and phrases:
L´evy process, exponential functional, general-ized Ornstein-Uhlenbeck process.
Contents ν . . . . . . . . . . . . . . . . . . . . . . 53 Estimation of the L´evy triplet . . . . . . . . . . . . . . . . . . . . . . . 63.1 Estimation of the Laplace exponent . . . . . . . . . . . . . . . . 63.2 Estimation of a and c . . . . . . . . . . . . . . . . . . . . . . . . 73.3 Recovering the L´evy measure ν . . . . . . . . . . . . . . . . . . . 104 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Theoretical study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 Appendix. Additional proofs . . . . . . . . . . . . . . . . . . . . . . . 21References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ∗ This research was partially supported by the Deutsche Forschungsgemeinschaft throughthe SFB 823 “Statistical modelling of nonlinear dynamic processes” and by Laboratory forStructural Methods of Data Analysis in Predictive Modeling, MIPT, RF government grant,ag. 11.G34.31.0073. 1 a r X i v : . [ s t a t . O T ] D ec .Belomestny and V.Panov/Statistical inference for exponential functionals
1. Introduction
For a L´evy process ξ = ( ξ t ) t ≥ , the exponential functional of ξ is defined by A t = (cid:90) t e − ξ s ds, where t ∈ (0 , ∞ ). The main object of this research is the terminal value A ∞ := lim t →∞ A t = (cid:90) ∞ e − ξ s ds, (1)which often (and everywhere in this paper) is called also an exponential func-tional of ξ . The integral A ∞ naturally arises in a wide variety of financial ap-plications as an invariant distribution of the process V t = e − ξ t (cid:18) V + (cid:90) t e ξ s − ds (cid:19) , (2)see Carmona, Petit, Yor [10]. For instance, the process (2) determines the volatil-ity process in the COGARCH (COntinious Generalized AutoRegressive Condi-tionally Heteroscedastic) model introduced by Kl¨uppelberg et al. [17]. Note that V t is in fact a partial case of the generalized Ornstein-Uhlenbeck (GOU) process.A comprehensive study of the GOU model is given in the dissertation by Behme[2]. A ∞ appears in finance also in other contexts, for instance, in pricing of Asianoptions, see the monograph by Yor [30] and the references given by Carmona,Petit, Yor [10]. As for other fields of applications, A ∞ plays a crucial role instudying the carousel systems (see Litvak and Adan [21], Litvak and Zwet [22]),self-similar fragmentations (see Bertoin and Yor [8]), and information transmis-sion problems (especially TCP/IP protocol, see Guillemin, Robert and Zwart[15]). For the detailed discussion of the physical interpretations, we refer toComtet, Monthus and Yor [11] and the dissertation by Monthus [24].Denote the L´evy triplet of the process ξ t by ( c, σ, ν ), i.e., ξ t = ct + σW t + T t , (3)where T t is a pure jump process with L´evy measure ν . The finiteness conditionstands that the integral A ∞ is finite if and only if ξ t → + ∞ as t → + ∞ , seeMaulik and Zwart [23] for the proof and Erickson and Maller [14] for some ex-tensions of this result. Therefore, the integral A ∞ is finite if the process ξ t is anynon-degenerated subordinator, i.e., any non-decreasing L´evy process, or, equiv-alently, any non-negative L´evy process. Nevertheless, the finiteness condition isfulfilled for other processes also, e.g., for ξ t = − N t + 2 λt , where N t is a Poissonprocess with intensity λ .In this paper, we mainly focus on the case when ξ is a subordinator withfinite L´evy measure. In terms of the L´evy triplet, this means that c > σ = 0, .Belomestny and V.Panov/Statistical inference for exponential functionals ν ( IR − ) = 0 and moreover a := ν ( IR + ) < ∞ . Suppose that the process (2) isobserved in the time points 0 = t < t < ... < t n . Taking into account that theprocess V t is a Markov process, and assuming that V has an invariant distribu-tion determined by A ∞ , we conclude that V t , ..., V t n have also the distributionof A ∞ . The main goal of this research is to statistically infer on the L´evy tripletof ξ from the observations V t , ..., V t n . More precisely, we will pursue the fol-lowing two aims: (1) to estimate the drift term c and the parameter a ; (2) toestimate the L´evy measure ν .To the best of our knowledge, the statistical inference for exponential func-tionals of L´evy processes has not been previously considered in the literature.However, some distributional properties of the exponential functionals are well-known, e.g., the integro - differential equation by Carmona, Petit, Yor [9]. Forthe overview of theoretical results, we refer to the survey by Bertoin and Yor[8]. One distribution property, the recursive formula for the moments of A ∞ ,gives rise to the approach presented in our paper. This result stands that E (cid:2) A s − ∞ (cid:3) = ψ ( s ) s E [ A s ∞ ] , (4)where ψ ( s ) is a Laplace exponent of the process ξ , i.e., ψ ( s ) := − log E (cid:2) e − sξ (cid:3) , and complex s is taken from the areaΥ := (cid:110) s ∈ C : 0 < Re( s ) < θ (cid:111) with θ := sup (cid:8) z ≥ E [ e − zξ ] ≤ (cid:9) . (5)The recursive formula (4) firstly appear for real s in the paper by Maulik andZwart [23]. The complete proof for complex s was given recently by Kuznetsov,Pardo and Savov [20]. If ξ t is a subordinator, what is the case under our setup,the parameter θ is equal to infinity.The idea of the procedure for solving the first task (estimation of a and c ) is toinfer on the parameters of the process ξ from its Laplace exponent. First, makinguse of (4), we estimate the Laplace exponent ψ ( s ) at the points s = u + i v ∈ Υ,where u is fixed and v varies on the equidistant grid between εV n and V n (with ε > V n → ∞ as n → ∞ ) Afterwards, we take into account that ψ ( u + i v ) = a + c ( u + i v ) − F ¯ ν ( − v ) , u, v ∈ IR, (6)where ¯ ν ( dx ) := e − ux ν ( dx ), and F ¯ ν ( v ) stands for the Fourier transform of themeasure ¯ ν , i.e., F ¯ ν ( v ) := (cid:82) IR + e i vx ¯ ν ( dx ) . It is worth mentioning that F ¯ ν ( v ) → v → ∞ , and therefore taking the real and imaginary parts of the left andright hand sides of (6), we are able to consequently estimate the parameters c and a .With no doubt, the second aim (complete recovering of the L´evy measure)is the most challenging task. Since the estimates of the parameters c and a are already obtained, we can estimate by (6) the Fourier transform F ¯ ν ( v ) for v taken from the equidistant grid [ − V n , V n ]. The last step of this procedure,estimation of the L´evy measure ν , is based on the inverse Fourier transformformula, and reveals the main reason for using the complex numbers in our .Belomestny and V.Panov/Statistical inference for exponential functionals approach. In fact, one can estimate the function ψ ( · ) in real points and thenestimate the Laplace transform of the measure ν by the regression arguments. Inthis case, estimation of ν demands the inverse Laplace transform, which is givenby Bromwich integral and therefore is in fact much more involved in comparisonwith the inverse Fourier transform.The paper is organized as follows. In the next section, we introduce the as-sumptions on this model and give some examples. We formulate the algorithmsfor estimation the Laplace exponent ψ ( s ) (Section 3.1), the parameters a and c (Section 3.2), and the L´evy measure ν (Section 3.3). Next, we provide some nu-merical examples in Section 4 and analyze the convergence rates of the proposedalgorithms in Section 5. Appendix contains some related results and additionalproofs.
2. Assumptions on the model
In this article, we restrict our attention to the case when the following set ofassumptions is fulfilled: (A1) (cid:40) c ≥ , σ = 0 ,ν ( IR − ) = 0 , a := ν ( IR + ) < ∞ . This set in particularly yields that the process ξ has finite variation, i.e., (cid:90) IR + ( x ∧ ν ( dx ) < ∞ , (7)and therefore ξ is a non-decreasing L´evy processes, i.e., a subordinator. Thedetailed discussion of the subordination theory as well as various examples ofsuch processes (Gamma, Poisson, tempered stable, inverse Gaussian, Meixnerprocesses, etc.), are given in [1], [7], [12], [26], [27].Note that in the case of subordinators, the truncation function in the L´evy-Khinchine formula can be omitted, and therefore the characteristic exponent of ξ is equal to ψ e ( s ) = log E (cid:2) e i sξ (cid:3) = i cs + (cid:90) ∞ (cid:0) e i sx − (cid:1) ν ( dx ) . (8)Later on, we use a Laplace exponent of ξ , which is defined by ψ ( s ) := − log E (cid:2) e − sξ (cid:3) = − ψ e (i s ) , and under the assumption (A1) is equal to ψ ( s ) = cs + (cid:90) ∞ (cid:0) − e − sx (cid:1) ν ( dx ) (9)= cs + s (cid:90) ∞ e − sx ν ( x, + ∞ ) dx. (10) .Belomestny and V.Panov/Statistical inference for exponential functionals Some examples can be found in Section 4. In the sequel, we use the fact thatthe function ψ ( · ) is bounded from above on the set Υ by | ψ ( s ) | ≤ c | s | + (cid:90) ∞ (cid:16) e − Re( s ) x (cid:17) ν ( dx ) ≤ c (cid:113) θ + Im ( s ) + 2 a, and hence the asymptotic behavior of the function ψ ( s ) is given by | ψ ( s ) | = O (cid:16) Im( s ) (cid:17) , Im( s ) → + ∞ . (11) ν First, we assume the following asymptotic behavior of the Mellin transform ofthe integral A ∞ : (A2) (cid:12)(cid:12) E (cid:2) A u ◦ +i v ∞ (cid:3)(cid:12)(cid:12) (cid:16) exp {− γ | v |} , as | v | → ∞ with some γ > u ◦ > ν ( dx ) := e − u ◦ x ν ( dx ) : (A3) (cid:13)(cid:13) ¯ ν ( r ) (cid:13)(cid:13) L ∞ ( IR ) ≤ C for some positive r and C .It is worth noting that there is an (indirect) relation between the assumptions(A2) and (A3). In fact, joint consideration of (4) and (6) yields that F ¯ ν ( − v ) = a + c ( u ◦ + i v ) − ( u ◦ + i v ) m ( u ◦ + i v ) m (( u ◦ + 1) + i v ) , u, v ∈ IR, where m ( s ) := E (cid:2) A s − ∞ (cid:3) is the Mellin transform. Example 1.
For instance, the set of assumptions (A1) - (A3) fulfills for theclass of L´evy processes with c = 0 and L´evy density in the form ν ( x ) = I x> M (cid:88) j =1 m j (cid:88) k =1 α jk x k − e − ρ j x with M, m j ∈ N , ρ j > α jk >
0. In fact, assumption (A1) and (A3) obviouslyhold; assumption (A2) is checked in [19] for any positive u ◦ (p. 658, the proofof Theorem 1). Example 2.
Next, we provide an example of the L´evy process which doesn’tpossess the property (A2). Consider a subordinator T with drift c > ν ( x ) = ab exp {− bx } I { x > } , a, b > , which we describe in details in Section 4. The exponential functional of thisprocess has a density k ( x ) = C x b (1 − cx ) ( a/c ) − I { < x < /c } .Belomestny and V.Panov/Statistical inference for exponential functionals with some C >
0, see [9]. In other words, the exponential functional has adistribution B ( α + 1 , β + 1) /c , where B stands for Beta distribution with pa-rameters α = b and β = a/c −
1. The Mellin transform of the function k ( x ) inthe half -plane Re( s ) > − α is given by m ( s ) = C ( α, β ) c s Γ( α + s )Γ( α + β + 1 + s ) , where C ( α, β ) >
0, see Table 1 from [13]. Taking into account the followingasymptotical behavior of the Gamma function | Γ( u + i v ) | = exp (cid:26) − π v + (cid:18) u − (cid:19) ln v + O (1) (cid:27) , v → ∞ , see (9) from [19], we conclude that the exponential functional of the process T has a polynomial decay of the Mellin transform. More precisely, | m ( s ) | (cid:16) C v − a/c − with some C >
3. Estimation of the L´evy triplet
In the sequel, suppose that the process (2) is observed at the time points 0 = t < t < ... < t n . Assuming that V has a stationary invariant distribution, weget that the values A ∞ ,k := V t k , k = 1 ..n have the distribution of the integral A ∞ . The first step of the estimation procedure is to construct the estimate of thefunction ψ ( s ) in the complex points s = u + i v , where u is fixed and v varies.The reason for such choice of s is clear from the further steps of the algorithm.The estimator of ψ ( s ) is based on a recursive formula for the s − th (complex)moment of A ∞ : E (cid:2) A s − ∞ (cid:3) = ψ ( s ) s E [ A s ∞ ] , (12)In [9], this formula is proved for real positive s such that ψ ( s ) > E [ A s ∞ ] < ∞ . The case of infinite mathematical expectations is carefully discussed in [23].The case of complex s is considered in [20], where one can find also somegeneralizations of the formula (12) for integrals with respect to the Brownianmotion with drift. In particular, applying Theorem 2 from [20], we get that(12) holds for any s ∈ Υ. In the situation when ξ t is a subordinator, the set Υcoincides with the positive half-plane (equivalently, the parameter θ is equal toinfinity), because it follows from (10) that E (cid:2) e − sξ (cid:3) = − ψ ( − s ) = − cs − s (cid:90) IR + e sx ν ( x, + ∞ ) dx < , s > . .Belomestny and V.Panov/Statistical inference for exponential functionals Motivated by (12), we now present the the first two steps in the estimationprocedure. Let the values α , ..., α M compose the equidistant grid with the step∆ > ε, ε > V n tends to infinity. First,we estimate A s ∞ for s = u + i α m V n , m = 1 ..M and s = ( u −
1) + i v m , m = 1 ..M where u := u ◦ ∈ ( − , θ ) satisfies the assumptions (A2) and (A3). Theoreticalstudies (see Section 5) show that the optimal choice is V n = κ log( n ) with κ < / (2 γ ), provided that the Assumptions (A1)-(A3) hold. The estimator of A s ∞ is defined by (cid:98) E n [ A s ∞ ] = 1 n n (cid:88) k =1 A s ∞ ,k . (13)Next, we define an estimate of ψ ( · ) at the points ( u + i α m V n ) byˆ ψ n ( u + i α m V n ) = ( u + i α m V n ) (cid:98) E n (cid:104) A ( u − α m V n ∞ (cid:105)(cid:98) E n (cid:104) A u +i α m V n ∞ (cid:105) , m = 1 ..M. (14)The performance of this estimator is later shown in Section 4, see in particularlyFigures 1 and 3. The quality of ˆ ψ n ( · ) is theoretically studied in Theorem 5.1,which stands that under the assumptions (A1) and (A2) and the following con-dition on V n Λ n := V n exp { cV n } (cid:112) log V n = o (cid:18)(cid:114) n log( n ) (cid:19) , n → ∞ , it holds for n large enough P (cid:40) sup v ∈ [ εV n ,V n ] (cid:12)(cid:12)(cid:12) ˆ ψ n ( u + i v ) − ψ ( u + i v ) (cid:12)(cid:12)(cid:12) ≤ β Λ n (cid:114) log( n ) n (cid:41) > − αn − − δ , (15)with some positive α , β and δ . a and c In Section 2.1, we present the representation (9) for the Laplace exponent ofthe process ξ . Substituting now the complex argument z = u + i v , we get ψ ( u + i v ) = c ( u + i v ) − (cid:90) IR + e − i vx ¯ ν ( dx ) + (cid:90) IR + ν ( dx )= c ( u + i v ) − F ¯ ν ( − v ) + a, u, v ∈ IR, (16)where a := (cid:82) IR + ν ( dx ) and ¯ ν ( dx ) := e − ux ν ( dx ). The general idea of the pro-cedure described below is to estimate the Laplace exponent ψ ( · ) at the points s = u + i v , where u is fixed at v varies (see Section 3.1), and afterwards to use(16) for consequent estimation of the parameters. .Belomestny and V.Panov/Statistical inference for exponential functionals Taking imaginary and real of both hand sides in (16), we getIm ψ ( u + i v ) = cv − Im F ¯ ν ( − v ) , (17)Re ψ ( u + i v ) = cu − Re F ¯ ν ( − v ) + a. (18)By Riemann - Lebesque lemma, F ¯ ν ( − v ) → v → + ∞ , see, e.g., [16]; notethat the rates of this convergence are assumed in (A3). Therefore, looking at(17), we conclude that Im ψ ( u + i v ) is a (asymptotically) linear in v function,and the parameter c can be interpreted a slope parameter. Next, from (18), itfollows that Re ψ ( u + i v ) tends to ( cu + a ) as v → + ∞ . These observations leadto the following optimization problems˜ c n := arg min c (cid:90) IR + w n ( v ) (cid:16) Im ˆ ψ n ( u + i v ) − cv (cid:17) dv (19)˜ a n := arg min a (cid:90) IR + w n ( v ) (cid:16) Re ˆ ψ n ( u + i v ) − ˜ c n u − a (cid:17) dv, (20)where the weighting function is chosen in the form w n ( v ) = w ( v/V n ) /V n withan integrable non-negative function w ( · ) supported on [ ε, V n , we can rewrite (19) as follows:˜ c n := arg min c (cid:90) ε w ( α ) (cid:16) Im ˆ ψ n ( u + i αV n ) − cαV n (cid:17) dv ˜ a n := arg min a (cid:90) ε w ( α ) (cid:16) Re ˆ ψ n ( u + i αV n ) − ˜ c n u − a (cid:17) dv, In practice, we first get the estimates of the Laplace exponent at the points s = u + i α m V n (see above) and define an estimate of the parameter c byˆ c n := arg min c M (cid:88) m =1 w ( α m ) (cid:16) Im ˆ ψ n ( u + i α m V n ) − cα m V n (cid:17) (21)= (cid:80) Mm =1 w ( α m ) α m Im ˆ ψ n ( u + i α m V n ) V n · (cid:80) Mm =1 w ( α m ) α m . (22)Afterwards, we estimate the parameter a byˆ a n := arg min a M (cid:88) m =1 w ( α m ) (cid:16) Re ˆ ψ n ( u + i α m V n ) − ˆ c n u − a (cid:17) (23)= (cid:80) Mm =1 w ( α m ) Re ˆ ψ n ( u + i α m V n ) (cid:80) Mm =1 w ( α m ) − ˆ c n u. (24)We show empirical and theoretical properties of the estimators ˆ a n and ˆ c n below, see Figure 2 and Theorem 5.3. Similar to (15), we prove that under the .Belomestny and V.Panov/Statistical inference for exponential functionals choice V n = κ log( n ) with κ < / (2 γ ), it holds P (cid:40) | ˜ c n − c | ≤ ζ log − ( r +2) ( n ) (cid:41) > − αn − − δ , and P (cid:40) | ˜ a n − a | ≤ ζ log − ( r +1) ( n ) (cid:41) > − αn − − δ , with ζ , ζ >
0, and α, δ introduced above. Constants γ and r involved in thisstatement are comming from assumptions (A2) and (A3) resp. Moreover, weprove Theorem 5.4, which stands that this rate for c is optimal one in the class A of the models satisfying the assumptions (A1) - (A3). More precisely, weshow that lim n →∞ inf ˜ c ∗ n sup A P (cid:40) | ˜ c ∗ n − c | ≥ ζ log − ( r +2) ( n ) (cid:41) > , where ζ < ζ is some positive constant, the supremum is taken over all modelsfrom A , and infimum - over all possible estimates of the parameter c .We summarize the steps discussed above in the following algorithm. .Belomestny and V.Panov/Statistical inference for exponential functionals Algorithm 1: Estimation of a and c Data : n observations A ∞ , , ..., A ∞ ,n of the integral A ∞ = (cid:82) IR + exp {− ξ s } ds ,where ξ = ( ξ t ) t ≥ is a L´evy process with unknown L´evy triplet( c, , ν ).Take V n = κ log( n ) with κ < / (2 γ ), fix ε ∈ (0 ,
1) and u > − α , ..., α M on the equidistant grid on the set [ ε, w ( · ) ≥ ε, v m,n := α m V n .1. Estimate A s ∞ for s = u j + i v m,n , m = 1 ..M, where u = u and u = u − (cid:98) E n (cid:2) A u j +i v m,n ∞ (cid:3) = 1 n n (cid:88) k =1 A u j +i v m,n ∞ ,k , m = 1 ..M, j = 1 , .
2. Estimate ψ ( u + i v m,n ) byˆ ψ n ( u + i v m,n ) = ( u + i v m,n ) (cid:98) E n (cid:104) A ( u − v m,n ∞ (cid:105)(cid:98) E n (cid:104) A u +i v m,n ∞ (cid:105) , m = 1 ..M.
3. Estimate c by the solution of the optimization problem (21),which is explicitly given byˆ c n := (cid:80) Mm =1 w ( α m ) α m Im ˆ ψ n ( u + i v m,n ) V n · (cid:80) Mm =1 w ( α m ) α m .
4. Estimate a by the solution of the optimization problem (23),which is explicitly given byˆ a n := (cid:80) Mm =1 w ( α m ) Re ˆ ψ n ( u + i v m,n ) (cid:80) Mm =1 w ( α m ) − ˆ c n u. ν As the result of the algorithm described below, we obtain the estimates ˆ c n andˆ a n of the parameters c and a . In this subsection, we present the algorithm forestimation the L´evy measure ν .First, we take points s = u + i α m V n , where α m , m = 1 ..M , belong to theinterval [ − , ψ n ( s ) remains the same asin Section 3.1. Next, looking at (16), we define an estimate F ¯ ν ( − v ) for v = .Belomestny and V.Panov/Statistical inference for exponential functionals α m V n , m = 1 ..M byˆ F ¯ ν ( − v ) = − ˆ ψ n ( u + i v ) + ˆ c n ( u + i v ) + ˆ a n . (25)The last step is to recover the measure ν from the estimator of the Fouriertransform of the measure ¯ ν . Motivated by the inverse Fourier transform formula,we propose the following nonparametric estimator of the measure ν :˜ ν ( x ) = 12 π e ux (cid:90) IR e i vx ˆ F ¯ ν ( − v ) K ( vh n ) dv, (26)where K is a regularizing kernel supported on [ − ,
1] and h n is a sequence ofbandwidths which tends to 0 as n → ∞ . The formal description of the algorithmis given below. Algorithm 2: Estimation of ν Data : n observations A ∞ , , ..., A ∞ ,n of the integral A ∞ = (cid:82) IR + exp {− ξ s } ds ,where ξ = ( ξ t ) t ≥ is a L´evy process with unknown L´evy triplet( c, , ν ). The estimates ˆ a n and ˆ c n are described in Algorithm 1.Take the values α , ..., α M on the equidistant grid on the set [ − , v m,n := α m V n .Define a regularizing kernel K supported on [ − , h .1-2 The first two steps coincide with given in Algorithm 1.3. Estimate F ¯ ν ( − v m,n ) for ¯ ν ( dx ) = e − ux ν ( dx ) byˆ F ¯ ν ( − v m,n ) = − ˆ ψ n ( u + i v m,n ) + ˆ c n · ( u + i v m,n ) + ˆ a n , m = 1 ..M.
4. Estimate ν byˆ ν ( x ) = e ux ∆2 π M (cid:88) m =1 e i v m,n x ˆ F ¯ ν ( − v m,n ) K ( v m,n h ) . Some theoretical and practical aspects of this algorithm are discussed inSections 4 and 5.
Remark 3.1.
It is a worth mentioning that the estimation algorithms 1 and2 can be applied to more general situation when the process T t is a differencebetween two subordinators, i.e., T t = T + t + T − t , where T + and T − are theprocesses of finite variation with L´evy measures ν + and ν − concentrated on IR + and IR − resp. In fact, in this case, the formula (16) still holds with ν ( dx ) = II { x > } ν + ( dx ) + II { x < } ν − ( dx ) . .Belomestny and V.Panov/Statistical inference for exponential functionals Therefore, the consequent estimation of c , a and the Fourier transform of themeasure e − ux ν ( dx ) , as well as the estimation of ν are still possible.Theoretical results under the assumptions (A2) and (A3) remain the same.The example from Section 2.2 can be naturally extended to ν ( x ) = I x> M (cid:88) j =1 m j (cid:88) k =1 α jk x k − e − ρ j x + I x< M (cid:88) j =1 ˜ m j (cid:88) k =1 ˜ α jk x k − e − ˜ ρ j x with M, ˜ M , m j , ˜ m j ∈ N , ρ j , ˜ ρ j > , α jk , ˜ αjk > . Note that Assumption (A2)is already checked in Theorem 1 from [19].
4. Simulation studyExample 1.
Consider the subordinator T t with the L´evy density ν ( x ) = ab exp {− bx } I { x > } , a, b > . (27)For this subordinator, the integral A ∞ is finite for any σ , see [9]. The Laplaceexponent of ξ is given by ψ ( z ) = z (cid:18) c − σ z + ab + z (cid:19) . (28)As for the distribution properties of A ∞ , the density function of A ∞ satisfiesthe following differential equation − σ x k (cid:48)(cid:48) ( x ) + (cid:20)(cid:18) σ − b ) + c (cid:19) x − (cid:21) k (cid:48) ( x )+ (cid:20) (1 − b ) (cid:18) σ c (cid:19) − a + bx (cid:21) k ( x ) = 0 , (29)see [9]. Some typical situations are given below:1. In the case c = 0 , σ = 0 (pure jump process), this equation has a solution k ( x ) = Cx b e − ax I { x > } , (30)and therefore A ∞ d G ( b + 1 , a ), where G ( α, β ) is a Gamma distributionwith shape parameter α and rate β .2. If c > , σ = 0 (pure jump process with drift), then k ( x ) = Cx b (1 − cx ) ( a/c ) − I { < x < /c } . (31)In this situation A ∞ d B ( b +1 , a/c ) /c , where B ( α, β ) is a Beta - distribution. .Belomestny and V.Panov/Statistical inference for exponential functionals −30 −10 10 30 . . . . . . Im(s) Re −30 −10 10 30 − − Im(s) Im −30 −10 10 30 Im(s)
Abs
Fig 1 . Plots of theoretical (blue dashed) and empirical (red solid) Laplace exponents for Exam-ple 1. Graphs present real, imaginary parts and absolute values. In spite of visual distinctionin the real parts, the difference between theoretical and empirical Laplace exponents is quitesmall.
3. In the case c (cid:54) = 0 , σ (cid:54) = 0, the equation (29) also allows for the closed formsolutions. Assuming for simplicity σ / c = − ( b + 1), we get thesolution of (29) in the following form: k ( x ) = C x b − / exp (cid:26) x (cid:27) I µ (cid:18) x (cid:19) , (32)where we denote by I µ the modified Bessel function of the first kind, µ = (cid:112) a + 1 /
4, and the constant c is later chosen to guarantee the condition (cid:82) ∞ k ( x ) dx = 1.For the numerical study, we assume that the data follows the model (1) wherethe process ξ t is defined by (3) with c = 1 . σ = 0, and the subordinator T t hasa L´evy density in the form (27) with a = 0 . b = 0 .
2. The values of the integral A ∞ are simulated from the Beta-distribution, see (31).On the first step, we estimate A s ∞ for s = u + i v with u = 29 and u = 30 and v from the equidistant grid between −
30 and 30. Next, we estimate the Laplaceexponent by the formula (14). One can visually compare the proposed estimatorand the theoretical value ( c + a/ ( b + s )) ∗ s looking at Figure 1.Estimation of the parameters c and a is provided by (22) and (25) resp. Theboxplots of this estimates are presented on Figure 2. Example 2.
Consider the compound Poisson process ξ t = − log q (cid:32) N t (cid:88) k =1 η k (cid:33) , .Belomestny and V.Panov/Statistical inference for exponential functionals . . . c . . . . . a Fig 2 . Boxplots for the estimates of c and a for different values of n based on 25 simulationruns. where q ∈ (0 ,
1) is fixed, N t is a Poisson process with intensity λ and η k arei.i.d. random variables with a distribution (cid:32)L. It is a worth mentioning that theintegral A ∞ allows the representation A ∞ = (cid:90) ∞ q − ξ t dt = ∞ (cid:88) n =0 q − n ( T n +1 − T n ) , where T n is the jump time T n = inf { t : N t = n } . Note that if η k takes onlypositive values then − ξ t is a subordinator. For the overview of the properties ofthe integral A ∞ in the particular case (cid:32)L ≡ ξ t is a Poisson process upto a constant), we refer to [8].Fix some positive α and consider the case when (cid:32)L is the standard Normaldistribution truncated on the interval ( α, + ∞ ). The density function of (cid:32)L isgiven by p (cid:32)L( x ) = p ( x ) / (1 − F ( α )) , where p ( · ) and F ( · ) are pdf and cdf of the standard Normal distribution. In thiscase, the Laplace exponent of ξ t is equal to ψ ( s ) = λ (cid:34) − − F ( α + (log q ) s )1 − F ( α ) exp (cid:40) − (log q ) s (cid:41)(cid:35) , where the function F ( · ) in the complex point z can be calculated from the errorfunction: F ( z ) := 12 (cid:18) erf (cid:18) z √ (cid:19) + 1 (cid:19) , where erf( z ) = 2 √ π (cid:90) z e − s ds. .Belomestny and V.Panov/Statistical inference for exponential functionals −4 0 2 4 . . . . v X Re −4 0 2 4 − . − . . . . v X Im −4 0 2 4 . . . . . v X Abs
Fig 3 . Plots of theoretical (blue dashed) and empirical (red solid) Laplace exponents forExample 2. Graphs present real, imaginary and absolute values. For v ∈ [ − , the curvesare visually indistinguishable. In this example, we aim to estimate the L´evy measure of the process ξ t , whichis equal to ν ( dx ) = λ − F ( α ) p ( x ) I { x > α } dx. For the numerical study, we take q = 0 . , α = 0 .
1, and λ = 1. First, we estimatethe Laplace exponent by (14). The quality of estimation at the complex points s = u + i v with u = 1 and v ∈ [ − ,
5] can be visually checked on Figure 3.Next, we proceed with the estimation of the Fourier transform of the measure¯ ν ( x ) := e − ux ν ( x ) of the L´evy measure by applying (25). For the last step of theAlgorithm 2, reconstruction of the L´evy measure by (26), we follow [4] and takethe so-called flat-top kernel, which is defined as follows: K ( x ) = , | x | ≤ . , exp (cid:16) − e − / ( | x |− . −| x | (cid:17) , . < | x | < , , | x | ≥ . The quality of the resulted estimation is given on Figure 4.
5. Theoretical studyTheorem 5.1.
Consider the model (1) with L´evy process ξ in the form (3) satisfying the assumptions (A1) - (A3). Let the sequence V n tend to ∞ andmoreover satisfy the assumption Λ n := V n exp { γV n } (cid:112) log V n = o (cid:18)(cid:114) n log( n ) (cid:19) , n → ∞ , (33) .Belomestny and V.Panov/Statistical inference for exponential functionals . . . . z Y Re − . − . − . − z Y Im Fig 4 . Plots of the L´evy measure (blue dashed line) and its estimate (red solid) depicted forreal (left) and imaginary (right) parts. Note that the values on the right plot are quite small. where the constant γ is introduced in (A2). Then there exists a set W n such that P {W n } > − αn − − δ (with some positive α and δ ) and W n ⊂ (cid:40) sup v ∈ [ εV n ,V n ] (cid:12)(cid:12)(cid:12) ˆ ψ n ( u + i v ) − ψ ( u + i v ) (cid:12)(cid:12)(cid:12) ≤ β Λ n (cid:114) log( n ) n (cid:41) (34) where β > and u = u ◦ was introduced in (A2). Remark 5.2.
The condition (33) fulfills for instance for V n = κ log( n ) with κ < / (2 γ ) .Proof. Denote J ( s ) := (cid:16)(cid:98) E n [ A s ∞ ] − E [ A s ∞ ] (cid:17) / E [ A s ∞ ] , where s = u + i v . In this notation, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ψ ( s ) − ˆ ψ n ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s (cid:32) E (cid:2) A s − ∞ (cid:3) E [ A s ∞ ] − (cid:98) E n (cid:2) A s − ∞ (cid:3)(cid:98) E n [ A s ∞ ] (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s E (cid:2) A s − ∞ (cid:3) E [ A s ∞ ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J ( s ) − J ( s − J ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (35)By (12), the first term is equal to | ψ ( s ) | , and therefore by (11) it is bounded by C Im( s ) for Im( s ) large enough with some C >
0. As for the second term, wefirstly note that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J ( s ) − J ( s − J ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | J ( s ) | + | J ( s − | − | J ( s ) | . .Belomestny and V.Panov/Statistical inference for exponential functionals The aim of the further proof is to show that the right hand side in the lastinequality is bounded by (cid:112) log( n ) /n on a probability set with desired properties. Proposition 6.2 yields that there exists such set W n of probability masslarger than 1 − αn − − δ , such that it holds on this setsup s : Im( s ) ∈ I n (cid:12)(cid:12)(cid:12)(cid:98) E n [ A s ∞ ] − E [ A s ∞ ] (cid:12)(cid:12)(cid:12) (cid:46) (cid:112) log( V n ) log( n ) /n, n → ∞ , (36)where α and δ are positive, and I n := [ εV n , V n ]. In fact, direct application ofProposition 6.2 with a weighting function w ∗ ( x ) := log − / ( e + | x | ) givessup v ∈ I n (cid:12)(cid:12)(cid:12)(cid:98) E n (cid:2) A u +i v ∞ (cid:3) − E (cid:2) A u +i v ∞ (cid:3)(cid:12)(cid:12)(cid:12) ≤ sup v ∈ I n (cid:20)(cid:12)(cid:12)(cid:12) w ∗ ( v )inf x ∈ I n w ∗ ( x ) (cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:16)(cid:98) E n (cid:2) A u +i v ∞ (cid:3) − E (cid:2) A u +i v ∞ (cid:3)(cid:17)(cid:12)(cid:12)(cid:12)(cid:105) ≤ (cid:112) log ( e + V n ) · sup v ∈ I n (cid:12)(cid:12)(cid:12) w ∗ ( v ) (cid:16)(cid:98) E n (cid:2) A u +i v ∞ (cid:3) − E (cid:2) A u +i v ∞ (cid:3)(cid:17)(cid:12)(cid:12)(cid:12) (cid:46) (cid:112) log( V n ) log( n ) /n, n → ∞ . Formula (36) in particularly means the following inequlity holds on the set W n sup s : Im( s ) ∈ I n | J ( s ) | (cid:46) exp { γV n } (cid:112) log( V n ) log( n ) /n n → ∞ , . (37)It is worth mentioning that under the assumption (33),sup s : Im( s ) ∈ I n | J ( s ) | → n → ∞ . (38)Substituting (37) into (35) and taking into account (11) and (38), we arrive atthe following bound for the quality of the estimate ˆ ψ n ( s ): (cid:12)(cid:12)(cid:12) ψ ( s ) − ˆ ψ n ( s ) (cid:12)(cid:12)(cid:12) (cid:46) V n exp { γV n } (cid:112) log( V n ) log( n ) /n, which holds on the set W n . This observation completes the proof. Theorem 5.3.
Consider the setup of Theorem 5.1 and take V n = κ log( n ) with κ < / (2 γ ) . Then it holds W n ⊂ (cid:40) | ˜ c n − c | ≤ ζ log r +2 ( n ) (cid:41) and W n ⊂ (cid:40) | ˜ a n − a | ≤ ζ log r +1 ( n ) . (cid:41) , (39) where s is introduced in (A3), the set W n is defined in Theorem 5.1, and ζ , ζ > . .Belomestny and V.Panov/Statistical inference for exponential functionals Proof. First note that the estimate (19) can be rewriten as˜ c n = (cid:90) ∞ w ∗ n ( v ) Im ˆ ψ n ( u + i v ) dv, where w ∗ n ( v ) = w n ( v ) v (cid:82) w n ( y ) y dy = 1 V n w ∗ (cid:18) vV n (cid:19) , where w ∗ ( x ) = ( w ( x ) x ) / (cid:0)(cid:82) w ( y ) y dy (cid:1) . Next, consider the following “theoreticalcounterpart” of the estimate ˜ c n :¯ c n := (cid:90) ∞ w ∗ n ( v ) Im ψ ( u + i v ) dv, and note that | ˆ c n − c | ≤ | ˆ c n − ¯ c n | + | ¯ c n − c | . (40)The first summand in the right hand side of (40) is bounded on the set W n for n large enough: | ˆ c n − ¯ c n | ≤ A Λ n (cid:114) log( n ) n V n , where A := V n β (cid:12)(cid:12)(cid:12)(cid:12) (cid:82) w ∗ n ( v ) vdv (cid:82) w ∗ n ( v ) v dv (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:82) ε w ∗ ( v ) vdv (cid:82) ε w ∗ ( v ) v dv (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (41)doesn’t depend on n . As for the second term, using (cid:82) w ∗ n ( v ) vdv = 1, we get | ¯ c n − c | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∞ w ∗ n ( v ) (cid:104) Im ˆ ψ n ( u + i v ) dv − cv (cid:105) dv (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∞ w ∗ n ( v ) Im F ¯ ν ( − v ) dv (cid:12)(cid:12)(cid:12)(cid:12) . Applying Lemma 6.3 with w ∗ n ( v ) = V − n w ∗ ( v/V n ), we get (cid:12)(cid:12)(cid:12)(cid:90) ∞ w ∗ n ( v ) F ¯ ν ( v ) dv (cid:12)(cid:12)(cid:12) (cid:46) V − ( r +2) n , n → ∞ . (42)Substituting (41) and (42) into (40), and bearing in mind our choice of V n ,we complete the proof of the first embedding in (39). Without limitations we can assume that (cid:82) IR + w n ( v ) dv = (cid:82) ε w ( v ) dv = 1.The second embedding directly follows from Theorem 5.1 and the first part of .Belomestny and V.Panov/Statistical inference for exponential functionals this proof, because | ˜ a n − a | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:34)(cid:90) IR + w n ( v ) Re ˆ ψ n ( u + i v ) dv − ˜ c n u (cid:35) − (cid:34)(cid:90) IR + w n ( v ) (cid:16) Re ψ ( u + i v ) + Re F ¯ ν ( − v ) (cid:17) dv − cu (cid:35)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | ˜ c n − c | u + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) IR + w n ( v ) (cid:16) Re ˆ ψ n ( u + i v ) − Re ψ ( u + i v ) (cid:17) dv (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) IR + w n ( v ) Re F ¯ ν ( − v ) dv (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) ζ log r +2 ( n ) + β Λ n (cid:114) log( n ) n + λ log r +1 ( n ) (cid:46) r +1 ( n ) , n → ∞ , where λ >
0. Note that here we use the inequality (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) IR + w n ( v ) Re F ¯ ν ( − v ) dv (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) log − ( r +1) ( n ) , which follows by applying Lemma 6.3 to w n ( v ) = V − n w n ( v/V n ). This completesthe proof. Theorem 5.4.
Let A be a set of functions that satisfy assumptions (A1) -(A3). Then it holds lim n →∞ inf ˜ c ∗ n sup A P (cid:40) | ˜ c ∗ n − c | ≥ ζ log − ( r +2) ( n ) (cid:41) > , where ζ is some positive constant, the supremum is taken over all models from A , and infimum - over all possible estimates of the parameter c .Proof. We follow the general reduction scheme, which can be found in [18] and[28]. Consider a class of L´evy processes A that satisfies the assumptions (A1)-(A3). There exist two L´evy process ξ and ξ from A , having L´evy triplets( c , , ν ), ( c , , ν ), Laplace exponents φ , φ , exponential functionals withdensities p , p and Mellin transforms M , M , such that it holds simultaneously1. the L´evy triplets are related by the following identities: c − c = 2 δ, ν ( x ) − ν ( x ) = 2 δK (cid:48) h ( x ) , (43)where δ > K h ( x ) = h − K (cid:0) h − x (cid:1) for any x ∈ IR and some h >
0, and K ∈ L ( C ) satisfy F K ( z ) = − z with Re( z ) ∈ [ − ,
1] and polynomicaldecay | F K ( z ) | (cid:46) | Re ( z ) | − η as | z | → ∞ . .Belomestny and V.Panov/Statistical inference for exponential functionals
2. the density of one of the functionals, say the first one, decays at mostpolynomially, i.e., there exists m ∈ N such that p ( x ) (cid:38) (1 + x ) − m , x → + ∞ . M ( s ) and M ( s ) coincide on the lines s = u ( k ) + i v, k = 1 , , for u (1) =3 / u (2) = m + 3 /
2, and any v . Moreover, the asymptotics of the Mellintransforms along these lines is given by (A2), i.e., | M j ( u ( k ) + i v ) | (cid:16) exp {− γ ( k ) | v |} , as v → ∞ , with some γ ( k ) > , k = 1 , , j = 0 , . Let the exponential functionals of these L´evy processes have distribution laws P and P . χ n (1 |
0) := χ ( P ⊗ n | P ⊗ n ) ≤ exp (cid:8) nχ ( P | P ) (cid:9) − , see Lemma 5.5 from [5]. The aim is to show that there exist a constant C > χ n (1 | < C ; after that the desired result will immediately follow, seePart 2 and especially Theorem 2.2 from [28].Our choice of the models leads to the following estimate of the chi-squareddistance between P and P : χ (1 |
0) := χ ( P | P ) = (cid:90) IR + ( p ( x ) − p ( x )) p ( x ) (cid:46) (cid:90) IR + (1 + x m ) ( p ( x ) − p ( x )) dx. (44)By Lemma 6.4 we get that χ (1 | (cid:46) ∆(0) + ∆( m ) , where ∆( · ) := (cid:90) ∞−∞ | M ( · + 1 / v ) − M ( · + 1 / v ) | dv. (45)Note that by (4) and our assumptions, M ( s − − M ( s −
1) = φ ( s ) − φ ( s ) s M ( s ) , (46)where s = u ( k ) + i v , k = 1 ,
2. By our choice of the L´evy measures (43) and therepresentation of the Laplace exponent (9), we get φ ( s ) − φ ( s ) = ( c − c ) s + (cid:90) IR + (cid:0) − e − sx (cid:1) ν ( dx ) − (cid:90) IR + (cid:0) − e − sx (cid:1) ν ( dx )= 2 δs + (cid:90) IR + [ ν ( dx ) − ν ( dx )] − [ F ν (i s ) − F ν (i s )]= 2 δs + 2 δ (cid:90) IR + K (cid:48) h ( x ) dx − δ F K (cid:48) h (i s ) .Belomestny and V.Panov/Statistical inference for exponential functionals Next, we take into account that F K (cid:48) h ( y ) = i y F K h ( y ) = i yF K ( yh ) for any y ∈ C .Therefore (cid:82) IR + K (cid:48) h ( x ) dx = F K (cid:48) h (0) = 0 and moreover φ ( s ) − φ ( s ) = 2 δs (1 + F K (i sh )) . (47)Substituting (47) into (46), we arrive at M ( s − − M ( s −
1) = 2 δ (1 + F K (i sh )) M ( s ) , and therefore∆( · ) = δ (cid:90) IR (cid:12)(cid:12)(cid:12) F K (cid:16)(cid:16) − v + i u ( k ) (cid:17) h (cid:17)(cid:12)(cid:12)(cid:12) ∗ (cid:12)(cid:12)(cid:12) M (cid:16) u ( k ) + i v (cid:17)(cid:12)(cid:12)(cid:12) dv, where k = 1 if · = 0 and k = 2 if · = m . By our assumptions on the kernel K ,we get ∆( · ) (cid:46) δ (cid:90) | v | > /h e − γ ( k ) | v | dv = δγ ( k ) e − γ ( k ) /h , and therefore χ (1 | (cid:46) δγ ∗ e − γ ∗ /h , with γ ∗ := min (cid:110) γ (1) , γ (2) (cid:111) . If we choose δ = h s +2 and h = log − ( n ) γ ∗ / (1 + ε ) for any (small) ε >
0, the χ - divergence is bounded by χ (1 |
0) = ( γ ∗ ) s +2 (1 + ε ) s +2 log − ( s +2) ( n ) n ε (cid:46) log( C + 1) n for any C > n large enough. Therefore, χ n (1 | ≤ exp { nχ (1 | } − ≤ C, and the statement of the theorem follows.
6. Appendix. Additional proofsLemma 6.1 (Exponential inequalities for dependent sequences) . Let ( G k , k ≥ be a sequence of centered real-valued random variables on the probability space (Ω , F , P ) . Assume that1. G k is a strongly mixing sequence with the mixing coefficients satisfying α G ( n ) ≤ ¯ α exp {− ¯ α n } , n ≥ , ¯ α > , ¯ α >
0; (48) sup k ≥ | G k | ≤ M a.s. for some positive M ; .Belomestny and V.Panov/Statistical inference for exponential functionals
3. the quantities ρ k := E (cid:104) G k | G k | ε ) (cid:105) , k = 1 , , . . . , are finite for all k with some small ε > .Then there is a positive constant C depending on ¯ α := (¯ α , ¯ α ) such that P (cid:40) n (cid:88) k =1 G k ≥ β (cid:41) ≤ exp (cid:20) − C β nv + M + M β log n (cid:21) . for all β > and n ≥ , where v ≤ sup k E [ G k ] + C sup k ρ k with C > .Proof. The proof directly follows from Theorem A.1 and Corollary A.2 from[6].The next result gives the uniform probabilistic inequality for the empiricalprocess. This result is an analogue of Proposition A.3 from [6], which gives theuniform inequality for the case when u = 0 (see below). For similar results ini.i.d. case, see [25]. Proposition 6.2.
Let Z j , j = 1 , . . . , n, be a stationary sequence of randomvariables. Define ϕ n ( v ) := 1 n n (cid:88) j =1 exp { ( u + i v ) Z j } , where u ∈ IR + is fixed and v ∈ IR varies. Let ϕ ( v ) be a characteristic functionof the corresponding stationary distribution. Let also w be a positive monotonedecreasing Lipschitz function on R + such that < w ( z ) ≤ (cid:112) log( e + | z | ) , z ∈ R . (49) Suppose that the following assumptions hold: (A1) random variables e Z j possess finite absolute moments of order p > . (A2) Z j is a strongly mixing sequence with the mixing coefficients satisfying α Z ( n ) ≤ ¯ α exp {− ¯ α n } , n ≥ , ¯ α > , ¯ α > . (50) Then there are δ (cid:48) > and ζ > , such that the inequality P (cid:26)(cid:114) n log n (cid:107) ϕ n − ϕ (cid:107) L ∞ ( R ,w ) > ζ (cid:27) ≤ Bζ − p n − − δ (cid:48) . (51) holds for any ζ > ζ and some positive constant B not depending on ζ and n. .Belomestny and V.Panov/Statistical inference for exponential functionals Proof.
Denote W n ( v ) := w ( v ) n n (cid:88) j =1 (cid:16) e ( u +i v ) Z j I (cid:8) e uZ j < Ξ n (cid:9) − E (cid:104) e ( u +i v ) Z I (cid:8) e uZ < Ξ n (cid:9)(cid:105)(cid:17) , W n ( v ) := w ( v ) n n (cid:88) j =1 (cid:16) e ( u +i v ) Z j I (cid:8) e uZ j ≥ Ξ n (cid:9) − E (cid:104) e ( u +i v ) Z I (cid:8) e uZ ≥ Ξ n (cid:9)(cid:105)(cid:17) , where Z is a random variable with stationary distribution of Z j . The main ideaof the proof is to show that P (cid:40) |W n ( v ) | > ζ (cid:114) log nn (cid:41) ≤ (cid:101) B ζ − p n − − δ (cid:48) , (52) P (cid:40) |W n ( v ) | > ζ (cid:114) log nn (cid:41) ≤ (cid:101) B ζ − p n − − δ (cid:48) , (53)with Ξ n = ... and some positive (cid:101) B and (cid:101) B . Step 1.
The aim of the first step is to show (52). The proof follows the samelines as the proof of Proposition A.3 from [6].
Consider the sequence A k = e k , k ∈ N and cover each interval [ − A k , A k ]by M k = ( (cid:98) A k /γ (cid:99) + 1) disjoint small intervals Λ k, , . . . , Λ k,M k of the length γ. Let v k, , . . . , v k,M k be the centers of these intervals. We have for any natural K > k =1 ,...,K sup A k − < | v |≤ A k |W n ( v ) | ≤ max k =1 ,...,K max ≤ m ≤ M k sup v ∈ Λ k,m |W n ( v ) − W n ( v k,m ) | + max k =1 ,...,K max (cid:110) ≤ m ≤ M k : | v k,m | >A k − (cid:111) |W n ( v k,m ) | . Hence for any positive λ , P (cid:32) max k =1 ,...,K sup A k − < | v |≤ A k |W n ( v ) | > λ (cid:33) ≤ P (cid:32) sup | v − v | <γ |W n ( v ) − W n ( v ) | > λ/ (cid:33) + K (cid:88) k =1 (cid:88)(cid:110) ≤ m ≤ M k : | v k,m | >A k − (cid:111) P ( |W n ( v k,m ) | > λ/ . (54)The aim of the next two steps is to get the upper bounds for the summandsin the right hand side, where λ is taken in the form λ = ζ (cid:112) (log n ) /n witharbitrary large enough ζ . .Belomestny and V.Panov/Statistical inference for exponential functionals We proceed with the first summand in (54). It holds for any v , v ∈ R |W n ( v ) − W n ( v ) | ≤ | w ( v ) − w ( v ) | × max v (cid:12)(cid:12)(cid:12)(cid:12) W n ( v ) w ( v ) (cid:12)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12) W n ( v ) w ( v ) − W n ( v ) w ( v ) (cid:12)(cid:12)(cid:12)(cid:12) × max v [ w ( v )] ≤ n | w ( v ) − w ( v ) | + 1 n n (cid:88) j =1 (cid:104)(cid:12)(cid:12)(cid:12) e ( u +i v ) Z j − e ( u +i v ) Z j (cid:12)(cid:12)(cid:12) I (cid:8) e uZ j < Ξ n (cid:9)(cid:105) + (cid:12)(cid:12)(cid:12) E (cid:104)(cid:16) e ( u +i v ) Z − e ( u +i v ) Z (cid:17) I (cid:8) e uZ < Ξ n (cid:9)(cid:105)(cid:12)(cid:12)(cid:12) ≤ | v − v | Ξ n L w + 1 n n (cid:88) j =1 | Z j | + E | Z | , (55)where L ω is the Lipschitz constant of w and Z is a random variable distributedby the stationary law of the sequence { Z j } . Next, the Markov inequality implies P n n (cid:88) j =1 (cid:104) | Z j | − E | Z | (cid:105) > c ≤ c − p n − p E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (cid:88) j =1 (cid:104) | Z j | − E | Z | (cid:105)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p for any c > . Using now Yokoyama inequality [29] and taking into accountthe assumptions of the continuity of moments of Z j and the assumption 1 fromLemma 6.1, we get E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (cid:88) j =1 (cid:104) | Z j | − E | Z | (cid:105)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p ≤ C p (¯ α ) n p/ , where C p (¯ α ) is some constant depending on ¯ α = (¯ α , ¯ α ) and p . Returning toour choice of γ and λ , which in particularly yields that γ = λ/ζ = (cid:112) (log n ) /n, we obtain from (55) P (cid:110) sup | v − v | <γ |W n ( v ) − W n ( v ) | > λ/ (cid:111) ≤ P n n (cid:88) j =1 (cid:104) | Z j | − E | Z | (cid:105) > ζ n − L w − E | Z | ≤ B c p (¯ α ) (cid:16) ζ/ (2Ξ n ) − L w − E | Z | (cid:17) − p n − p/ ≤ B ζ − p Ξ pn n − p/ with some constants B , B not depending on ζ and n, provided ζ is largeenough. Now we turn to the second term on the right-hand side of (54). ApplyingLemma 6.1 with G k = n Re (cid:2) W n ( u k,m ) (cid:3) and β = nλ , we get P (cid:0) | Re (cid:2) W n ( v k,m ) (cid:3) | > λ/ (cid:1) ≤ K , .Belomestny and V.Panov/Statistical inference for exponential functionals where K := exp (cid:32) − B λ nB Ξ n w ( A k − ) log ε ) (Ξ n w ( A k − )) + λ log ( n )Ξ n w ( A k − ) (cid:33) with some constants B and B depending only on the characteristics of theprocess Z . Similarly, applying the same result with G k = n Im (cid:2) W n ( u k,m ) (cid:3) , weconclude that P (cid:0) | Im (cid:2) W n ( v k,m ) (cid:3) | > λ/ (cid:1) ≤ K , and therefore (cid:88) {| v k,m | >A k − } P ( |W n ( v k,m ) | > λ/ ≤ ( (cid:98) A k /γ (cid:99) + 1) K . Set now γ = (cid:112) (log n ) /n and λ = ζ (cid:112) (log n ) /n and note that under our choiceof Ξ n , Ξ n w ( A k − ) log ε ) (Ξ n w ( A k − )) (cid:38) λ log ( n )Ξ n w ( A k − ) . Therefore, (cid:88) {| v k,m | >A k − } P ( |W n ( v k,m ) | > λ/ (cid:46) A k (cid:114) n log( n ) exp (cid:32) − Bζ log( n ) w ( A k − )Ξ n log ε ) ( w ( A k − )) (cid:33) , n → ∞ with some constant B > . Fix θ >
Bθ > (cid:88) {| v k,m | >A k − } P ( |W n ( v k,m ) | > λ/ (cid:46) (cid:114) n log( n ) exp (cid:110) k − θB ( k − − B ( k − ζ (log n/ Ξ n ) − θ ) (cid:111) (cid:46) (cid:114) n log( n ) e k (1 − θB ) e − B ( k − ζ (log n/ Ξ n ) − θ ) . Since ζ (log n/ Ξ n ) > θ , we arrive at K (cid:88) k =2 (cid:88) {| v k,m | >A k − } P ( |W n ( v k,m ) | > λ/ (cid:46) (cid:114) n log( n ) e − B ( ζ (log n/ Ξ n ) − θ ) (cid:104) K (cid:88) k =2 e k (1 − θB ) (cid:105) (cid:46) log − / ( n ) exp (cid:110) − Bζ (log n/ Ξ n ) + log( n ) (cid:111) . .Belomestny and V.Panov/Statistical inference for exponential functionals Taking large enough ζ >
0, we get (52).
Step 2 . Now we are concentrated on (53). The idea of the proof given belowwas published in [3], Proposition 7.4.Consider the sequence R n ( v ) := 1 n n (cid:88) j =1 e ( u +i v ) Z j I (cid:8) e uZ j ≥ Ξ n (cid:9) . By the Markov inequality we get | E [ R n ( u )] | ≤ E (cid:2) e uZ j (cid:3) P (cid:8) e uZ j ≥ Ξ n (cid:9) ≤ Ξ − pn E (cid:2) e uZ j (cid:3) E (cid:2) e upZ j (cid:3) = o (cid:16)(cid:112) (log n ) /n (cid:17) Set ν k = 2 k , k ∈ , , ... , then it holds ∞ (cid:88) k =1 P (cid:110) max j =1 ..η k +1 e uZ j ≥ Ξ η k (cid:111) ≤ ∞ (cid:88) k =1 η k +1 P { e uZ ≥ Ξ η k }≤ E e puZ ∞ (cid:88) k =1 η k +1 Ξ − pη k < ∞ . By the Borel-Cantelli lemma, P (cid:110) max j =1 ..η k +1 e uZ j ≥ Ξ η k for infinitely many k (cid:111) = 0 . From here it follows that R n ( u ) − E R n ( u ) = o (cid:16)(cid:112) (log n ) /n (cid:17) . This completesthe proof. Lemma 6.3.
Let the measure ¯ ν be such that (cid:107) ¯ ν ( r ) (cid:107) ∞ ≤ C for some positive C , the weighting function w n admits the property w n = V − kn w ( v/V n ) for some k > and function w satisfying (cid:107)F w ( u ) /u r ( · ) (cid:107) L ≤ C with some C > . Then (cid:12)(cid:12)(cid:12)(cid:90) ∞ w n ( v ) F ¯ ν ( v ) dv (cid:12)(cid:12)(cid:12) (cid:46) V − ( r + k ) n , n → ∞ . Proof.
Following [5], we apply the Plancherel identity: (cid:12)(cid:12)(cid:12)(cid:90) ∞ w n ( v ) F ¯ ν ( v ) dv (cid:12)(cid:12)(cid:12) = 2 π (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) IR ¯ ν ( r ) ( x ) F − w n ( · ) / (i · ) r ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ π V − ( r + k ) n (cid:107) ¯ ν ( r ) (cid:107) ∞ (cid:107)F w ( u ) /u r ( · ) (cid:107) L (cid:46) V − ( r + k ) n . .Belomestny and V.Panov/Statistical inference for exponential functionals Lemma 6.4 (analogue of the Parseval-Plancherel theorem for Mellin trans-form) . Let X and X be two L´evy process with expontional functionals thathave densities p and p , and Mellin transforms M and M , resp. For any b ∈ IR , it holds (cid:90) ∞ x b ( p ( x ) − p ( x )) dx = (2 π ) − / (cid:90) ∞−∞ | M ( b/ / v ) − M ( b/ / v ) | dv. References [1] Barndorff-Nielsen, Ole E. and Shiryaev, A.N.
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