Moment bounds and central limit theorems for Gaussian subordinated arrays

TL;DR

This paper develops new moment bounds and central limit theorems for sums of nonlinear functions of Gaussian arrays, extending previous results and providing Berry-Esseen bounds with applications to Gaussian process statistics.

Contribution

It introduces a general moment bound for Gaussian functionals and extends CLTs to non-stationary Gaussian arrays, including Berry-Esseen bounds and applications.

Findings

Established a general moment bound for Gaussian functionals.

Proved a CLT for nonlinear functionals of non-stationary Gaussian arrays.

Derived a Berry-Esseen bound for the CLT.

Abstract

A general moment bound for sums of products of Gaussian vector's functions extending the moment bound in Taqqu (1977, Lemma 4.5) is established. A general central limit theorem for triangular arrays of nonlinear functionals of multidimensional non-stationary Gaussian sequences is proved. This theorem extends the previous results of Breuer and Major (1981), Arcones (1994) and others. A Berry-Esseen-type bound in the above-mentioned central limit theorem is derived following Nourdin, Peccati and Podolskij (2011). Two applications of the above results are discussed. The first one refers to the asymptotic behavior of a roughness statistic for continuous-time Gaussian processes and the second one is a central limit theorem satisfied by long memory locally stationary process.