VMSTA Modern Stochastics: Theory and Applications 2351-6054 2351-6046 2351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania VMSTA04 10.15559/vmsta-2014.1.1.4 Research Article LAN property for discretely observed solutions to Lévy driven SDE’s IvanenkoD.ida@univ.kiev.uaa∗ KulikA.kulik.alex.m@gmail.comb Taras Shevchenko National University of Kyiv, Kyiv, Ukraine Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine Corresponding author. 2014 2762014113347 442014 3052014 562014 © 2014 The Author(s). Published by VTeX2014 Open access article under the CC BY license.

The LAN property is proved in the statistical model based on discrete-time observations of a solution to a Lévy driven SDE. The proof is based on a general sufficient condition for a statistical model based on discrete observations of a Markov process to possess the LAN property, and involves substantially the Malliavin calculus-based integral representations for derivatives of log-likelihood of the model.

 LAN property likelihood function Lévy driven SDE regular statistical experiment 60J75 60H07 62F12 Introduction

Consider a stochastic equation of the form dXtθ=aθ(Xtθ)dt+dZt, where {aθ(x),θ∈Θ,x∈R} is a measurable function, Θ=(θ1,θ2)⊂R is a parametric set. For a given θ∈Θ , assuming that the drift term aθ satisfies the standard local Lipschitz and linear growth conditions, Eq. (1) uniquely defines a Markov process X. The aim of this paper is to establish the local asymptotic normality property (LAN in the sequel) in a model, where the process X is discretely observed with a fixed time discretization value h>0 0$]]>, and a number of observations n→∞ .

The LAN property provides a convenient and powerful tool for establishing lower efficiency bounds in a statistical model, e.g. [6, 17, 18]. Such a property for statistical models, based on discrete observations of processes with Lévy noise, was studied mostly in the cases where the likelihood function (or, at least its “main part”) is explicit in a sense, e.g., [1, 2, 7, 12, 13]. In the above references the models are linear in the sense that the process under the observation is either a Lévy process, or a solution of a linear (Ornstein-Uhlenbeck type) SDE driven by a Lévy process. The general non-linear case remains unstudied to a great extent, and apparently the main reason for this is that the transition probability density of the observed Markov process in this case is highly implicit. In this paper we develop the tools convenient for proving the LAN property in the framework of discretely observed solutions to SDE’s with a Lévy noise. To make the exposition reasonably transparent we confine ourselves to a particular case of one-dimensional and one-parameter model and a fixed sample frequency h. The various extensions (general state space, multiparameter model, high frequency sampling, etc.) are possible, but we postpone their detailed analysis for a further research.

Our approach consists of two principal parts. On one hand, we design a general sufficient condition for a statistical model based on a discrete observations of a Markov process to possess the LAN property, see Theorem 1 below. This result extends the classical LeCam’s result about the LAN property for i.i.d. samples. It is closely related to [5, Theorem 13], but with some substantial differences in the basic assumptions which makes our result well designed to study a model based on observations of a Lévy driven SDE, see Remark 1 below. On the other hand, we exploit Malliavin calculus-based integral representations of derivatives of 1st and 2nd orders for the log-likelihood, which we have derived in our recent papers [11] and [10]. The combination of these two principal parts leads to a required LAN property. We note that in the diffusion setting with high frequency sampling a Malliavin calculus-based approach to the proof of the LAN property is developed in [4]. Our approach is substantially different. The changes are yielded by a non-diffusive structure of the noise.

The structure of the paper follows the two-stage scheme outlined above. First, we formulate in Section 2.1 (and prove in Section 3) a general sufficient condition for the LAN property in a Markov model. Then we formulate in Section 2.2 (and prove in Section 4) our main result about the LAN property for the discretely observed solution to a Lévy driven SDE; here the proof involves the Malliavin calculus-based integral representations of derivatives of the log-likelihood from [11] and [10].

 The main results LAN property for discretely observed Markov processes

Let X be a Markov process taking its values in a locally compact metric space X . The law of X is assumed to be dependent on a real-valued parameter θ; in what follows, we assume that the parametric set Θ is an interval (θ1,θ2)∈R . We denote by Pxθ the law of X with X0=x , which corresponds to the parameter value θ; the expectation w.r.t. Pxθ is denoted by Exθ . For a given h>0 0$]]>, we denote by Px,nθ the law w.r.t. Pxθ of the vector Xn={Xhk,k=1,…,n} for discrete time observations of X with the step h. Denote by En the statistical experiment generated by the sample Xn with X0=x , i.e. En=(Xn,B(Xn),Px,nθ,θ∈Θ); we refer to [8] for the notation and terminology. Our aim is to establish the LAN property for the sequence of experiments {En} .

Recall that the sequence of statistical experiments {En} (or, equivalently, the family {Px,nθ,θ∈Θ} ) is said to have the LAN property at the point θ0∈Θ as n→∞ if for some sequence r(n)>0,n≥1 0,n\ge 1$]]> and all u∈R Zn,θ0(u):=dPx,nθ0+r(n)udPx,nθ0(Xn)=exp{Δn(θ0)u−12u2+Ψn(u,θ0)}, with L(Δn(θ0)|Px,nθ0)⇒N(0,1),n→∞; Ψn(u,θ0)⟶Px,nθ00,n→∞.

In what follows we assume that X admits a transition probability density ph(θ;x,y) w.r.t. some σ-finite measure λ. Furthermore, we assume that the experiment E1 is regular; that is, for every x∈X

the function θ↦ph(θ;x,y) is continuous for λ-a.s. y∈X ;

the function ph(θ;x,·) is differentiable in L2(X,λ) ; that is, there exists qh(θ;x,·)∈L2(X,λ) such that ∫X(ph(θ+δ;x,y)−ph(θ;x,y)δ−qh(θ;x,y))2λ(dy)→0,δ→0;

the function qh(θ;x,·) is continuous in L2(X,λ) w.r.t. θ; that is, ∫X(qh(θ+δ;x,y)−qh(θ;x,y))2λ(dy)→0,δ→0.

Denote gh(θ,x,y)=2qh(θ;x,y)ph(θ;x,y); note that the function gh is well defined by the definition of qh and satisfies Exθgh(θ;x,Xh)=0 for every x∈R , θ∈Θ . Furthermore, denote In(θ)= ∑k=1nExθ(gh(θ;Xh(k−1),Xhk))2=4Exθ ∑k=1n∫X(qh(θ;Xh(k−1),y))2λ(dy). Assuming that the statistical experiment En is regular, the above integral is finite and defines the Fisher information for En .

We fix θ0∈Θ , and put r(n)=In−1/2(θ0) for large enough n, assuming that for those n one has In(θ0)>0 0$]]>.

Suppose the following.

Statistical experiment (2) is regular for every x∈X and n≥1 and In(θ0)>0 0$]]> for large enough n.

The sequence r(n) ∑j=1ngh(θ0;Xh(j−1),Xhj),n≥1 is asymptotically normal w.r.t. Pxθ0 with parameters (0,1) .

The sequence r2(n) ∑j=1ngh2(θ0;Xh(j−1),Xhj),n≥1 converges to 1 in Pxθ0 -probability.

There exists a constant p>2 2$]]> such that limn→∞rp(n)Exθ0 ∑j=1n|gh(θ0;Xh(j−1),Xhj)|p=0.

For every N>0 0$]]> limn→∞sup|v|<Nr2(n)Exθ0 ∑j=1n∫X(qh(θ0+r(n)v;Xh(j−1),y)−qh(θ0;Xh(j−1),y))2λ(dy)=0.

Then {Px,nθ,θ∈Θ} has the LAN property at the point θ0 .

The above theorem is closely related to [5, Theorem 13]. One important difference is that in [5] the main conditions are formulated in the terms of the functions ph(θ+t;x,y)/ph(θ;x,y)−1, while within our approach the main assumptions are imposed on the log-likelihood derivative gh(θ;x,y) , and can be verified efficiently e.g. in a model where X is defined by an SDE with jumps (see Section 2.2 below). Another important difference is that the whole approach in [5] is developed under the assumption that the log-likelihood function smoothly depends on the parameter θ. For a model where X is defined by an SDE with jumps, such an assumption may be very restrictive (see the detailed discussion in [11]). This is the reason why we use the assumption of regularity of the experiments instead. It is much milder and easily verifiable (see [11]).

Let us note briefly two possible extensions of the result above, can be obtained without any essential changes in the proof. We do not expose them here in details since they will not be used in the current paper.

The statement of Theorem 1 still holds true if h is allowed to depend on n with conditions 1–5 have been changed, respectively.

The statement of Theorem 1 still holds true if, instead of one θ0 , a sequence θn→θ0 is considered, with conditions 2–5 have been changed, respectively. Moreover, in that case relations (3) and (4) would still hold true if instead of the fixed u the sequence un→u is considered. That is, under the uniform version of conditions 2–5 the uniform asymptotic normality holds true (see [8, Definition 2.2]).

 LAN property for families of distributions of solutions to Lévy driven SDE’s

We assume that Z in the SDE (1) is a Lévy process without a diffusion component; that is, Zt=ct+∫0t∫|u|>1uν(ds,du)+∫0t∫|u|≤1uν˜(ds,du), 1}u\nu (ds,du)+{\int _{0}^{t}}\int _{|u|\le 1}u\tilde{\nu }(ds,du),\]]]> where ν is the Poisson point measure with the intensity measure dsμ(du) , and ν˜(ds,du)=ν(ds,du)−dsμ(du) is the respective compensated Poisson measure. In the sequel, we assume the Lévy measure μ to satisfy the following.

For some β>0 0$]]>, ∫|u|≥1u4+βμ(du)<∞;

For some u0>0 0$]]>, the restriction of μ on [−u0,u0] has a positive density m∈C2([−u0,0)∪(0,u0]) ;

There exists C0 such that |m′(u)|≤C0|u|−1m(u),|m″(u)|≤C0u−2m(u),|u|∈(0,u0];

(log1ε)−1μ({u:|u|≥ε})→∞,ε→0.

One particularly important class of Lévy processes satisfying H consists of tempered α-stable processes (see [21]), that arise naturally in models of turbulence [20], economical models of stochastic volatility [3], etc.

Denote by Ck,m(R×Θ) , k,m≥0 the class of functions f:R×Θ→R that have continuous derivatives, ∂i∂xi∂j∂θjf,i≤k,j≤m.

We assume the coefficient aθ(x) in Eq. (1) to satisfy the following.

a∈C3,2(R×Θ) has bounded derivatives ∂xa , ∂xx2a , ∂xθ2a , ∂xxx3a , ∂xθθ3a , ∂xxθ3a , ∂xxxθ4a and |aθ(x)|+|∂θaθ(x)|+|∂θθ2aθ(x)|≤C(1+|x|),θ∈Θ,x∈R.

For a given θ0∈Θ , there exists a neighbourhood (θ−,θ+)⊂Θ of θ0 such that lim sup|x|→∞aθ(x)x<0uniformly inθ∈(θ−,θ+).

It is proved in [11] that under conditions A(i) and H, the following properties hold:

the Markov process X given by (1) has a transition probability density ptθ w.r.t. the Lebesgue measure;

this density has a derivative ∂θptθ(x,y) , and the statistical experiment (2) is regular;

the function gtθ , given by (5) satisfies (6).

 Hence all the pre-requisites for Theorem 1, given in Section 2.1, are true with λ(dx)=dx (the Lebesgue measure).

Furthermore, under conditions A and H, for θ=θ0 the corresponding Markov process X is ergodic, i.e. there exists the unique invariant probability measure ϰinvθ0 for X. One can verify this easily, using conditions sufficient for ergodicity of solutions to Lévy driven SDE’s, given in [19] and [14]. Denote by {Xtst,θ0,t∈R} the corresponding stationary version of X; that is, a Markov process, defined on whole R , which has the same transition probabilities as X and one-dimensional distributions equal to ϰinvθ0 . Clearly, the existence of such a process, on a proper probability space, is guaranteed by the Kolmogorov consistency theorem. Denote σ2(θ0)=E(gh(θ0;X0st,θ0,Xhst,θ0))2=∫R∫R(gh(θ0;x,y))2ph(θ0;x,y)dyϰinvθ0(dx).

The following theorem performs the main result of this paper. Its proof is given in Section 4 below.

Let conditions A and H hold true and σ2(θ0)>0. 0.\]]]> Then the family {Px,nθ,θ∈Θ} possesses the LAN property at the point θ=θ0 .

 Proof of Theorem <xref rid="j_vmsta04_stat_001">1</xref>

The method of proof goes back to LeCam’s proof of the LAN property for i.i.d. samples, see e.g. Theorem II.1.1 and Theorem II.3.1 in [8]. In the Markov setting, the dependence in the observations leads to some additional technicalities (see e.g. (19)). The possible ways to overcome these additional difficulties can be found, in a slightly different setting, in the proof of [5, Theorem 13]. In order to keep the exposition transparent and self-contained, we prefer to give a complete proof of Theorem 1 rather than to give a chain of partly relevant references.

We divide the proof into several lemmas; in all the lemmas in this section we assume the conditions of Theorem 1 to be fulfilled. Values x,θ0 and u are fixed; we assume that n is large enough, so that θ0+r(n)u∈Θ . In order to simplify the notation below we write θ instead of θ0 .

Denote ζjnθ(u)=((ph(θ+r(n)u;Xh(j−1),Xhj)ph(θ;Xh(j−1),Xhj))1/2−1)I{ph(θ;Xh(j−1),Xhj)≠0}.

One has lim supn→∞ ∑j=1nExθ(ζjnθ(u))2≤14u2 and limn→∞ ∑j=1nExθ(ζjnθ(u)−12r(n)ugh(θ;Xh(j−1),Xhj))2=0.

By the regularity of E1 and the Cauchy inequality we have Exθ(ζjnθ(u)−12r(n)ugh(θ;Xh(j−1),Xhj))2=Exθ∫{y:phθ(z,y)≠0}(ph(θ+r(n)u;Xh(j−1),y)−ph(θ;Xh(j−1),y)−r(n)uqh(θ;Xh(j−1),y))2λ(dy)≤(r(n)u)2Exθ∫R(∫01qh(θ+r(n)uv,Xh(j−1),y)−qh(θ;Xh(j−1),y)dv)2λ(dy)≤(r(n)u)2Exθ∫Rλ(dy)∫01(qh(θ+r(n)uv;Xh(j−1),y)−qh(θ;Xh(j−1),y))2dv. This inequality and (9) yield (13). To deduce (12) from (13) recall an elementary inequality |AB|≤α2A2+12αB2,α>0, 0,\]]]> and write ζjnθ(u)=12r(n)ugh(θ;Xh(j−1),Xhj)+(ζjnθ(u)−12r(n)ugh(θ;Xh(j−1),Xhj))=:A+B. Then Exθ(ζjnθ(u))2≤(1+α)14u2r2(n)Exθ(gh(θ;Xh(j−1),Xhj))2+(1+12α)Exθ(ζjnθ(u)−12r(n)ugh(θ;Xh(j−1),Xhj))2. Recall that ∑j=1nExθ(gh(θ;Xh(j−1),Xhj))2=In(θ)=r−2(n), hence the above inequality and (13) lead to the bound lim supn→∞ ∑j=1nExθ(ζjnθ(u))2≤1+α4u2. Since α>0 0$]]> is arbitrary, this completes the proof.  □

One has ∑j=1n(ζjnθ(u))2→u24,n→∞ in Pxθ -probability.

By the Chebyshev inequality, Pxθ{| ∑j=1n(ζjnθ(u))2−14r2(n)u2 ∑j=1n(gh(θ;Xh(j−1),Xhj))2|>ε}≤1ε ∑j=1nExθ|(ζjnθ(u))2−14r2(n)u2(gh(θ;Xh(j−1),Xhj))2|=1ε ∑j=1nExθ|ζjnθ(u)−12r(n)ugh(θ;Xh(j−1),Xhj)|×|ζjnθ(u)+12r(n)ugh(θ;Xh(j−1),Xhj)| \varepsilon \bigg\}\\{} & \displaystyle \hspace{1em}\le \frac{1}{\varepsilon }{\sum \limits_{j=1}^{n}}{\mathsf{E}_{x}^{\theta }}\bigg|{\big({\zeta _{jn}^{\theta }}(u)\big)}^{2}-\frac{1}{4}{r}^{2}(n){u}^{2}{\big(g_{h}(\theta ;X_{h(j-1)},X_{hj})\big)}^{2}\bigg|\\{} & \displaystyle \hspace{1em}=\frac{1}{\varepsilon }{\sum \limits_{j=1}^{n}}{\mathsf{E}_{x}^{\theta }}\bigg|{\zeta _{jn}^{\theta }}(u)-\frac{1}{2}r(n)ug_{h}(\theta ;X_{h(j-1)},X_{hj})\bigg|\\{} & \displaystyle \hspace{2em}\times \bigg|{\zeta _{jn}^{\theta }}(u)+\frac{1}{2}r(n)ug_{h}(\theta ;X_{h(j-1)},X_{hj})\bigg|\end{array}\]]]> which by (14), for a given α>0 0$]]>, is dominated by 12αε ∑j=1nExθ(ζjnθ(u)−12r(n)ugh(θ;Xh(j−1),Xhj))2+α2ε ∑j=1nExθ(ζjnθ(u)+12r(n)ugh(θ;Xh(j−1),Xhj))2. By (13) the first item of this expression tends to zero as n→∞ . Furthermore, the Cauchy inequality together with (12) and (15) implies that for the second one the upper limit does not exceed lim supn→∞(αε ∑j=1nExθ(ζjnθ(u))2+αu22εr2(n) ∑j=1nExθ(gh(θ;Xh(j−1),Xhj))2)≤3αu22ε. Since α>0 0$]]> is arbitrary, this proves that the difference ∑j=1n(ζjnθ(u))2−14r2(n)u2 ∑j=1n(gh(θ;Xh(j−1),Xhj))2 tends to 0 in Pxθ -probability. Combined with the condition 3 of Theorem 1, this gives the required statement.  □

One has max1≤j≤n|ζjnθ(u)|→0,n→∞ in Pxθ -probability.

We have Pxθ{max1≤j≤n|ζjnθ(u)|>ε}≤ ∑j=1nPxθ{|ζjnθ(u)|>ε}≤ ∑j=1nPxθ{|ζjnθ(u)−12r(n)ugh(θ;Xh(j−1),Xhj)|>ε2}+ ∑j=1nPxθ{|gh(θ;Xh(j−1),Xhj)|>ε4r(n)|u|}. \varepsilon \Big\}& \displaystyle \le {\sum \limits_{j=1}^{n}}{\mathsf{P}_{x}^{\theta }}\big\{\big|{\zeta _{jn}^{\theta }}(u)\big|>\varepsilon \big\}\\{} & \displaystyle \le {\sum \limits_{j=1}^{n}}{\mathsf{P}_{x}^{\theta }}\bigg\{\bigg|{\zeta _{jn}^{\theta }}(u)-\frac{1}{2}r(n)ug_{h}(\theta ;X_{h(j-1)},X_{hj})\bigg|>\frac{\varepsilon }{2}\bigg\}\\{} & \displaystyle \hspace{1em}+{\sum \limits_{j=1}^{n}}{\mathsf{P}_{x}^{\theta }}\bigg\{\big|g_{h}(\theta ;X_{h(j-1)},X_{hj})\big|>\frac{\varepsilon }{4r(n)|u|}\bigg\}.\end{array}\]]]> The first sum in the r.h.s. of this inequality vanishes as n→∞ because of (13), the second sum vanishes because of the condition 4 of Theorem 1.  □

By Lemma 3 and Lemma 2, we have ∑j=1n|ζjnθ(u)|3→0,n→∞ in Pxθ -probability.

Because of the Markov structure of the sample, in addition to Lemma 2 we will need the following statement. Denote Fj=σ(Xhi,i≤j),Ex,jθ=Exθ[·|Fj].

One has ∑j=1nEx,j−1θ(ζjnθ(u))2→u24,n→∞ in Pxθ -probability.

Denote χjn=(ζjnθ(u))2−Ex,j−1θ(ζjnθ(u))2,Sn= ∑j=1nχjn, then by (16) it us enough to prove that Sn→0 in Pxθ -probability. Fix ε>0 0$]]>, and put χjnε=(ζjnθ(u))21|ζjnθ(u)|≤ε−Ex,j−1θ((ζjnθ(u))21|ζjnθ(u)|≤ε),Snε= ∑j=1nχjnε. By construction, {χjε,j=1,…,n} is a martingale difference, hence Exθ(Snε)2= ∑k=1nExθ(χjnε)2≤ ∑k=1nExθ(ζjnθ(u))41|ζjnθ(u)|≤ε≤ε2Exθ ∑k=1n(ζjnθ(u))2. Hence by (12) and the Cauchy inequality, lim supn→∞Exθ|Snε|≤ε|u|2

Now, let us estimate the difference Sn−Snε . Note that, using the first statement in Lemma 1, one can improve the statement of Lemma 2 and show that the convergence (16) holds true in L1(Pxθ) ; see e.g. Theorem A.I.4 in [8]. In particular, this means that the sequence ∑j=1n(ζjnθ(u))2,n≥1 is uniformly integrable. Hence, because by Lemma 3 the probabilities of the sets Ωnε={maxj≤n|ζjn|>ε} \varepsilon \Big\}\]]]> tend to zero as n→∞ , we have Exθ(1Ωnε ∑j=1n(ζjnθ(u))2)→0. One has χjn−χjnε=(ζjnθ(u))21|ζjnθ(u)|>ε−Ex,jθ(ζjnθ(u))21|ζjnθ(u)|>ε, \varepsilon }-{\mathsf{E}_{x,j}^{\theta }}{\big({\zeta _{jn}^{\theta }}(u)\big)}^{2}1_{|{\zeta _{jn}^{\theta }}(u)|>\varepsilon },\]]]> hence Exθ|Sn−Snε|≤2 ∑j=1nExθ(ζjnθ(u))21|ζjnθ(u)|>ε≤2Exθ(1Ωnε ∑j=1n(ζjnθ(u))2)→0. \varepsilon }\le 2{\mathsf{E}_{x}^{\theta }}\bigg(1_{{\varOmega _{n}^{\varepsilon }}}{\sum \limits_{j=1}^{n}}{\big({\zeta _{jn}^{\theta }}(u)\big)}^{2}\bigg)\to 0.\end{array}\]]]> Together with (20) this gives lim supn→∞Exθ|Sn|≤ε|u|2, which completes the proof because ε>0 0$]]> is arbitrary.  □

The final preparatory result we require is the following.

One has 2 ∑j=1nζjnθ(u)−r(n)u ∑j=1ngh(θ;Xh(j−1),Xhj)→−u24,n→∞ in Pxθ -probability.

We have the equality (ζjnθ(u))2=ph(θ+r(n)u;Xh(j−1),Xhj)ph(θ;Xh(j−1),Xhj)−1−2ζjnθ(u) valid Pxθ -a.s. Note that by the Markov property of X one has Ex,j−1θph(θ+r(n)u;Xh(j−1),Xhj)ph(θ;Xh(j−1),Xhj)=∫Xph(θ+r(n)u;Xh(j−1),y)ph(θ;Xh(j−1),y)ph(θ;Xh(j−1),y)λ(dy)=1; hence by Lemma 4 one has that ∑j=1nEx,j−1θζjnθ(u)→−u28 in Pxθ -probability. Therefore, what we have to prove in fact is that Vn:=2 ∑j=1n(ζjnθ(u)−Ex,j−1θζjnθ(u))−r(n)u ∑j=1ngh(θ;Xh(j−1),Xhj)→0 in Pxθ -probability. By (6) the sequence ζjnθ(u)−Ex,j−1θζjnθ(u)−12r(n)ugh(θ;Xh(j−1),Xhj),j=1,…,n is a martingale difference, hence ExθVn2≤4 ∑j=1nExθ(ζjnθ(u)−12r(n)ugh(θ;Xh(j−1),Xhj))2, which tends to zero as n→∞ by (13).  □

Now, we can finalize the proof of Theorem 1. Fix ε∈(0,1) and consider the sets Ωnε defined by (21); by Lemma 3 we have Pxθ(Ωnε)→0 . Using the Taylor expansion for the function log(1+x) , we obtain that there exist a constant Cε and random variables αjn such that |αjn|<Cε , for which the following identity holds true outside of the set  Ωnε : ∑j=1nlogph(θ+r(n)u;Xh(j−1),Xhj)ph(θ;Xh(j−1),Xhj)=2 ∑j=1nζjnθ(u)− ∑j=1n(ζjnθ(u))2+ ∑j=1nαjn|ζjnθ(u)|3.

Then by Lemma 2, Lemma 5, and Corollary 1 we have logZn,θ(u)= ∑j=1nlogph(θ+r(n)u;Xh(j−1),Xhj)ph(θ;Xh(j−1),Xhj)=r(n)u ∑j=1ngh(θ;Xh(j−1),Xhj)−u24−u24+Ψn, where Ψn→0 in Pxθ -probability. By the asymptotic normality condition 2, this completes the proof.  □

 Proof of Theorem <xref rid="j_vmsta04_stat_005">2</xref>

To prove Theorem 2 we verify the conditions of Theorem 1. While doing that, we use the constructions and results from our recent papers [11, 10].

The regularity property, required in condition 1 of Theorem 1, is already proved in [11]. To prove other claims, involved into the conditions of Theorem 1, we will use the following auxiliary result several times.

Under conditions A and H for every p∈(2,4+β) there exists a constant C such that for all x∈R , θ∈(θ−,θ+) and t≥0 Exθ|gh(θ;x,Xh)|p≤C(1+|x|)p,Exθ|Xt|p≤C(1+|x|p).

The first inequality is proved in Lemma 1 [11]. One can prove the second inequality, using a standard argument based on the Lyapunov condition for the function V(x)=|x|p ; e.g. Proposition 4.1 [14].  □

Let us outline briefly the subsequent argument. To prove conditions 2 and 3 of Theorem 1, we need in fact to prove a CLT and a LLN for the sums ∑j=1ngh(θ0;Xh(j−1),Xhj) . The way to do this is quite standard: one should prove first such limit theorems for the stationary version of the process X, and then derive the limit behaviour of these sums under Pxθ0 . In this last step, the ergodic rates for the process X, and therefore the assumption A(ii), are essential. In the first step, which concerns the stationary version of the process X, we will need the following moment bounds for the invariant measure ϰinvθ0 for the process X with θ=θ0 .

Recall (e.g. [14], Section 3.2) that one standard way to construct ϰinvθ0 is to take a weak limit point (as T→∞ ) for the family of Khas’minskii’s averages ϰTθ0(dy)=1T∫0TPxθ0(Xt∈dy)dt. Then, by the Fatou lemma, the second relation in (23) implies the following moment bounds for ϰinvθ0 .

For every p∈(2,4+β) , ∫R|y|pϰinvθ0(dy)<∞.

Everywhere below we assume conditions of Theorem 2 to hold true.

The sequence 1n ∑j=1ngh(θ0;Xh(j−1),Xhj),n≥1 is asymptotically normal w.r.t. Pxθ0 with parameters (0,σ2(θ0)) , where σ2(θ0) is defined in (11).

The idea of the proof is similar to the one of the proof of Theorem 3.3 [16]. Denote Qn(θ0,X)=1n ∑j=1ngh(θ0;Xh(j−1),Xhj). By Theorem 2.2 [19] (see also Theorem 1.2 [14]), the α-mixing coefficient α(t) for the stationary version Xst of the process X does not exceed C3e−C4t , where C3,C4 are some positive constants. Then by CLT for stationary sequences (Theorem 18.5.3 [9]), the first relation in (23), and (24) we have Qn(θ0,Xst,θ0)⇒N(0,σ˜2(θ0)),n→∞ with σ˜2(θ0)= ∑k=−∞+∞E(gh(θ0;X0st,θ0,Xhst,θ0)gh(θ;Xh(k−1)st,θ0,Xhkst,θ0)). Furthermore, under conditions of Theorem 2 there exists an exponential coupling for the process X; that is, a two-component process Y=(Y1,Y2) , possibly defined on another probability space, such that Y1 has the distribution Pxθ0 , Y2 has the same distribution with Xst,θ0 , and for all t>0 0$]]> P(Yt1≠Yt2)≤C1e−C2t with some constants C1 , C2 . The proof of this fact can be found in [15] (Theorem 2.2). Then for any Lipschitz continuous function f:R→R we have |Exθf(Qn(θ0,X))−Ef(Qn(θ0,Xst,θ0))|=|Ef(Qn(θ0,Y1))−Ef(Qn(θ0,Y2))|≤Lip(f)E|Qn(θ0,Y1)−Qn(θ0,Y2)|≤Lip(f)n ∑k=1nE|gh(θ0;Yh(k−1)1,Yhk1)−gh(θ0;Yh(k−1)2,Yhk2)|1(Yh(k−1)1,Yhk1)≠(Yh(k−1)2,Yhk2)≤2Lip(f)n ∑k=1n(E|gh(θ0;Yh(k−1)1,Yhk1)|p+E|gh(θ0;Yh(k−1)2,Yhk2)|p)1/p×(P(Yh(k−1)1≠Yh(k−1)2)+P(Yhk1≠Yhk2))1/q, where p,q>1 1$]]> are such that 1/p+1/q=1 . Since Y1 has the distribution Pxθ0 , by (23) we have for p∈n(2,4+β) E|gh(θ0;Yh(k−1)1,Yhk1)|p=Exθ0|gh(θ0;Xh(k−1),Xhk)|p≤CExθ0(1+|Xh(k−1)|p))≤C+C2(1+|x|p). Similarly, E|gh(θ0;Yh(k−1)2,Yhk2)|p=E|gh(θ0;Xh(k−1)st,θ0,Xhkst,θ0)|p≤CE(1+|Xh(k−1)st,θ0|p))=C+C∫R|y|pϰinvθ0(dy), and the constant in the right hand side is finite by Corollary 2. Hence (25) and (26) yield that Exθf(Qn(θ0,X))→Ef(ξ),n→∞,ξ∼N(0,σ˜2(θ0)) for every Lipschitz continuous function f:R→R . This means that the sequence Qn(θ0,X),n≥1 is asymptotically normal w.r.t. Pxθ0 with parameters (0,σ˜2(θ0)) .

To conclude the proof, it remains to show that σ˜2(θ0)=σ2(θ0) . This follows easily from (6) because, by the Markov property of Xst,θ0 , σ˜2(θ0)=σ2(θ0)+2 ∑k=1∞E(gh(θ0;X0st,θ0,Xhst,θ0)gh(θ;Xh(k−1)st,θ0,Xhkst,θ0))=σ2(θ0)+2 ∑k=1∞E[gh(θ;X0st,θ0,Xhst,θ0)(Exθgh(θ0;x,Xh))x=Xh(k−1)st,θ0].  □

Similarly, one can prove that 1n ∑j=1n(gh(θ0;Xh(j−1),Xhj))2→σ2(θ0),n→∞ in L1(Pxθ0) ; the argument is completely the same, with the CLT for a stationary sequence replaced by the Birkhoff-Khinchin ergodic theorem (we omit the details). Hence In(θ0)∼nσ2(θ0),r(n)∼1nσ(θ0),n→∞. This proves that conditions 2–4 of Theorem 1 hold true. Condition 1 of Theorem 1 also holds true: regularity property is already proved in [11], and positivity of In(θ) follows from (29).

Let us prove (9), which then would allow us to apply Theorem 1. It is proved in [10] that, under the conditions of Theorem 2, the function qh(θ,x,y) is L2 -differentiable w.r.t. θ, and ∂θqh=12(∂θgh)ph+14(gh)2ph. In addition, it is proved therein that for every γ∈[1,2+β/2) Exθ|∂θgh(θ;x,Xh)|γ≤C(1+|x|)γ.

Then Exθ∫R(qh(θ+r(n)v,Xh(j−1),y)−qh(θ,Xh(j−1),y))2dy≤r(n)vExθ∫Rdy∫0r(n)v(∂θqh(θ+s,Xh(j−1),y))2ds≤r(n)v4Exθ∫0r(n)vds∫R(∂θgh(θ+s;Xh(j−1),y)+12gh(θ+s;Xh(j−1),y)2)2×phs(Xh(j−1),y)dy≤Cr(n)2v2Exθ(1+(Xh(j−1))4); in the last inequality we have used (30) and the first relation in (23). Using the second relation in (23), we get then sup|v|<Nr(n)2Exθ ∑j=1nExθ∫R(qh(θ+r(n)v,Xh(j−1),y)−qh(θ,Xh(j−1),y))2dy≤CN2nr(n)4 with a constant C that depends only on x. This relation together with (29) completes the proof.

 Acknowledgements

The authors are deeply grateful to H. Masuda for a valuable bibliographic help and useful discussion, and for a referee for the attention to the paper and useful remarks.

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