Asymptotic independence of multiple Wiener-It\^{o} integrals and the resulting limit laws

TL;DR

This paper characterizes the asymptotic independence of multiple Wiener-Itô integrals and derives key limit theorems, including the fourth moment theorem and its multidimensional extension, with applications to Gaussian and non-Gaussian processes.

Contribution

It provides a new characterization of asymptotic independence for Wiener-Itô integrals and extends fundamental limit theorems to multivariate and non-Gaussian contexts.

Findings

Established asymptotic independence for blocks of Wiener-Itô integrals.

Derived the multidimensional extension of the fourth moment theorem.

Applied results to Gaussian time series and non-Gaussian chaoses.

Abstract

We characterize the asymptotic independence between blocks consisting of multiple Wiener-It\^{o} integrals. As a consequence of this characterization, we derive the celebrated fourth moment theorem of Nualart and Peccati, its multidimensional extension and other related results on the multivariate convergence of multiple Wiener-It\^{o} integrals, that involve Gaussian and non Gaussian limits. We give applications to the study of the asymptotic behavior of functions of short and long-range dependent stationary Gaussian time series and establish the asymptotic independence for discrete non-Gaussian chaoses.