A multiple stochastic integral criterion for almost sure limit theorems

TL;DR

This paper establishes a criterion based on kernels of multiple stochastic integrals to prove almost sure central limit theorems, with applications to fractional Brownian motion increments.

Contribution

It introduces a new kernel-based criterion for almost sure limit theorems involving multiple stochastic integrals, extending previous results.

Findings

Almost sure CLTs hold for normalized sums of Hermite polynomial increments of fractional Brownian motion.

The criterion applies to a broad class of multiple stochastic integrals.

Convergence in law to normal distribution is established under the new criterion.

Abstract

In this paper, we study almost sure central limit theorems for multiple stochastic integrals and provide a criterion based on the kernel of these multiple integrals. We apply our result to normalized partial sums of Hermite polynomials of increments of fractional Brownian motion. We obtain almost sure central limit theorems for these normalized sums when they converge in law to a normal distribution.