Abstract
In [1], a framework to prove almost sure central limit theorems for sequences $(G_n)$ belonging to the Wiener space was developed, with a particular emphasis of the case where $G_n$ takes the form of a multiple Wiener-Ito integral with respect to a given isonormal Gaussian process. In the present paper, we complement the study initiated in [1], by considering the more general situation where the sequence $(G_n)$ may not need to converge to a Gaussian distribution. As an application, we prove that partial sums of Hermite polynomials of increments of fractional Brownian motion satisfy an almost sure limit theorem in the long-range dependence case, a problem left open in [1].
Highlights
Introduction and main resultsLet us start with the following definition, which plays a pivotal role in the paper
In [1], a framework to prove almost sure central limit theorems for sequences (Gn) belonging to the Wiener space was developed, with a particular emphasis of the case where going to prove that (Gn) takes the form of a multiple Wiener-Itô integral with respect to a given isonormal Gaussian process
We prove that partial sums of Hermite polynomials of increments of fractional Brownian motion satisfy an almost sure limit theorem in the long-range dependence case, a problem left open in [1]
Summary
Let us start with the following definition, which plays a pivotal role in the paper. Definition 1.1. A sequence (Gn) of multiple Wiener-Itô integrals converging in law to some μ being given, our goal in this paper is to provide a meaningful set of conditions under which an ASLT holds true. (Gn) satisfies an ASLT with μ = μH,q the Hermite distribution with parameters H and q To conclude this introduction, we would like to stress that our Theorem 1.3 is a true extension of Theorem 1.2, in the sense that the latter can be obtained as a particular case of the former: see Section 5 for the details.
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