Cumulants on Wiener chaos: moderate deviations and the fourth moment theorem

TL;DR

This paper establishes moderate and large deviation principles for sequences within Wiener chaos, utilizing cumulant estimates, and applies these results to various stochastic processes and fields.

Contribution

It extends the fourth moment theorem to include moderate and large deviations, providing new cumulant-based estimates for Wiener chaos elements.

Findings

Derived moderate deviation principles for Wiener chaos sequences

Established deviation inequalities under fourth moment theorem conditions

Applied results to Brownian sheet integrals and Gaussian fields

Abstract

A moderate deviation principle as well as moderate and large deviation inequalities for a sequence of elements living inside a fixed Wiener chaos associated with an isonormal Gaussian process are shown. The conditions under which the results are derived coincide with those of the celebrated fourth moment theorem of Nualart and Peccati. The proofs rely on sharp estimates for cumulants. As applications, explosive integrals of a Brownian sheet, a discretized version of the quadratic variation of a fractional Brownian motion and the sample bispectrum of a spherical Gaussian random field are considered.