Almost sure limit theorems on Wiener chaos: the non-central case

TL;DR

This paper extends almost sure limit theorems on Wiener chaos to non-Gaussian cases, applying the results to Hermite polynomial sums of fractional Brownian motion increments with long-range dependence.

Contribution

It generalizes previous work by establishing almost sure limit theorems for non-Gaussian sequences in Wiener chaos, including long-range dependent processes.

Findings

Proved almost sure limit theorems for non-Gaussian Wiener chaos sequences.

Applied results to Hermite polynomial sums of fractional Brownian motion.

Addressed open problem for long-range dependence case.

Abstract

In \cite{BNT}, 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-It\^o integral with respect to a given isonormal Gaussian process. In the present paper, we complement the study initiated in \cite{BNT}, 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 \cite{BNT}.