Berry-Esséen bounds and almost sure CLT for quadratic variation of weighted fractional Brownian motion
© Shen et al.; licensee Springer 2013
Received: 24 October 2012
Accepted: 17 May 2013
Published: 3 June 2013
In this paper, using the recent results on Stein’s method combining with Malliavin calculus and the almost sure central limit theorem for sequences of functionals of general Gaussian fields developed by Nourdin and Peccati, we derive the explicit bounds for the Kolmogorov distance in the central limit theorem and obtain the almost sure central limit theorem for the quadratic variation of the weighted fractional Brownian motion.
MSC:60F05, 60G15, 60H07.
Self-similar stochastic processes with long range dependence are of practical interest in various applications, including econometrics, Internet traffic and hydrology. These are processes whose dependence on the time parameter t are self-similar in the sense that there exists a (self-similarity) parameter such that for any constant , and have the same distribution. These processes are often endowed with other distinctive properties.
The fractional Brownian motion (fBm for short) is the usual candidate to model phenomena in which the self-similarity property can be observed from the empirical data. The fBm is a suitable generalization of the standard Brownian motion, which exhibits long-range dependence, self-similarity and has stationary increments. Some surveys and complete literatures could be found in Biagini et al. , Hu , Mishura , Nualart . On the other hand, many authors have proposed to use more general self-similar Gaussian processes and random fields as stochastic models. Such applications have raised many interesting theoretical questions about self-similar Gaussian processes and fields in general. Therefore, some generalizations of the fBm such as bi-fractional Brownian motion, sub-fractional Brownian motion and the weighted fractional Brownian motion have been introduced. However, in contrast to the extensive studies on fBm, there has been little systematic investigation on other self-similar Gaussian processes. The main reason for this is the complexity of dependence structures for self-similar Gaussian processes which do not have stationary increments.
and , , , the ranges of values of a and b being and . The process also appeared in Bojdecki et al.  in a high density limit of occupation time fluctuations of the above mentioned particles system, where the initial Poisson configuration has finite intensity measure, with , , . Moreover, wfBm was first studied by Bojdecki et al. , and it is neither a semimartingale nor a Markov process unless , , so many powerful techniques from stochastic analysis are not available when dealing with . The wfBm has properties analogous to those of fBm (self-similarity, long-range dependence, Hölder paths). However, in comparison with fBm, the wfBm has non-stationary increments. On the other hand, Garzón  showed that for certain values of the parameters the weighted fractional Brownian sheets were obtained as limits in law of occupation time fluctuations of a stochastic particle model.
Stein’s method is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric (see Stein ). The Malliavin calculus is an infinite-dimensional differential calculus, involving operators defined on the class of functionals of a given Gaussian stochastic process. Recently, Nourdin and Peccati [10, 11] unveiled the surface of the deep connection between the Malliavin calculus and Stein’s method in order to derive explicit bounds in the Gaussian and gamma approximations of random variables in a fixed Wiener chaos of a general Gaussian process. In particular, such an approach implies that on any fixed Wiener chaos each one of the well-known distances (Kolmogorov, total variation, Wasserstein) generates the weak topology when the limit is a Gaussian random variable. Some important non-linear functionals of Gaussian processes can be written as multiple stochastic integrals. An important particular example is the quadratic variation which lies in the second Wiener chaos. In the case of fBm, Breuer and Major  proved the central limit theorem for the quadratic variation when the Hurst parameter , Breton and Nourdin  did it for the critical value . Park et al.  studied a central limit theorem of cross-variation related to the standard Brownian sheet and provided the exact Berry-Esséen bound on the Kolmogorov distance. Kim  studied a central limit theorem of the cross variation related to fractional Brownian sheet with Hurst parameter such that . On the other hand, Bercu et al.  studied the almost sure central limit theorems for sequences of functionals of general Gaussian fields. Tudor  proved the almost sure central limit theorem for the quadratic variation of the sub-fractional Brownian motion. Aazizi et al.  and Liu  studied the bi-fractional Brownian motion case, respectively.
Motivated by all these results, in the present work, we consider the explicit bounds for the Kolmogorov distance in the central limit theorem and almost sure central limit theorems (ASCLT in short) for the quadratic variation of wfBm. The above mentioned properties make wfBm a possible candidate for models which involve long-range dependence, self-similarity and non-stationarity. Therefore, it seems interesting to study the Berry-Esséen bounds and ASCLT for the quadratic variation of the wfBm.
This paper is organized as follows. Section 2 contains some preliminaries for the Malliavin calculus and main results: Theorem 2.1 and Theorem 2.2. The proofs of the main results are given in Section 3.
We will use and to denote positive and finite constants depending on a, b only which may not be the same in each occurrence.
2 Malliavin calculus on wfBm and main results
In this section, we present the basic elements of Gaussian analysis and the Malliavin calculus that are used in this paper. Some surveys and a complete list of literature could be found in Nualart .
Let ℋ be a real separable Hilbert space. For any , we denote to be the q th tensor product of ℋ and to be the associated q th symmetric tensor product. We write to indicate a centered isonormal Gaussian process over ℋ defined on some probability space . This means that X is a centered Gaussian family, whose covariance is given in terms of the scalar product of ℋ by .
where stands for the usual Kronecker symbol, for , and . Moreover, if , we have , where is the symmetrization of f.
that is, we identify r variables in f and g and integrate them out.
for . Clearly, the function is of class and it is the distribution function of an absolutely continuous positive measure with density belonging to for .
with . Notice that the elements of the Hilbert space ℋ may not be functions but distributions of negative order.
The following result due to Nourdin and Peccati (see Theorem 3.1 and Proposition 3.2 of ) is very useful and reduces the problem of the normal approximation of multiple stochastic integrals (more general of random variables which are Malliavin derivable) in the Kolmogorov distance (and other standard metrics) to the estimation of the variance of the Malliavin derivative of the multiple integral (of the random variable, respectively).
as n tends to infinity, for every bounded and continuous function, where .
The following convenient criterion which extends the convergence in law to almost surely convergence (in particular for ASCLT) is due to Ibragimov and Lifshits . It plays a crucial role in all of the sequel.
as n tends to infinity, for every bounded and continuous function.
Applying the above criterion to multiple stochastic integrals, Bercu et al.  proved the following ASCLT.
- (1)For every ,(2.6)
then satisfies the ASCLT.
The aim of the present work is to state and prove the following two results.
Theorem 2.2 If , then the sequence satisfies the almost sure central limit theorem.
3 Proof of the main results
if and only if .
for some positive constant .
holds. Then we finish the proof of this theorem by Proposition 2.1. □
To present a proof of Theorem 2.2, the following proposition is needed.
Proposition 3.1 If , then the sequence satisfies the almost sure central limit theorem.
where is a fractional Brownian motion with Hurst index . □
Now we can give the proof of Theorem 2.2.
Proof of Theorem 2.2 From Theorem 2.1, we know that satisfies the central limit theorem. Hence we only need to check the conditions (1) and (2) in Proposition 2.3. The rest of the proof could be proved along the line of the proof of Theorem 5.1 of Bercu et al.  (see also Theorem 4.2 of Tudor , Theorem 4.2 of Aazizi et al. , Theorem 2.2 of Liu ). For the sake of completeness, we give the main arguments of the proof.
We consider and separately.
Thus, the condition (1) in Proposition 2.3 is satisfied.
It follows from (3.9) that the condition (2) in Proposition 2.3 is satisfied.
It follows that the condition (2) in Proposition 2.3 is satisfied. □
The authors would like to thank anonymous earnest referee whose remarks and suggestions greatly improved the presentation of the paper. Guangjun Shen is partially supported by the National Natural Science Foundation of China (11271020), the Natural Science Foundation of Anhui Province (1208085MA11) and the Key Natural Science Foundation of Anhui Educational Committee (KJ2011A139, KJ2012ZD01). Litan Yan is partially supported by the National Natural Science Foundation of China (11171062), Innovation Program of Shanghai Municipal Education Commission (12ZZ063). Jing Cui is partially supported by the Natural Science Foundation of Anhui Province (1308085QA14), the Key Natural Science Foundation of Anhui Educational Committee (KJ2013A133) and the Philosophy and Social Science Planning Foundation of Anhui Province (AHSK11-12D128).
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