Strong asymptotic independence on Wiener chaos

TL;DR

This paper establishes a precise criterion for asymptotic independence of components in sequences of Wiener chaos random vectors, based on the decay of covariances of their squares, refining previous moment-based criteria.

Contribution

It introduces a new inequality for multiple Wiener-Itô integrals and refines existing criteria for asymptotic independence in Wiener chaos.

Findings

Asymptotic independence characterized by covariance decay.

New inequality for Wiener-Itô integrals.

Refinement of moment-based independence criteria.

Abstract

Let $F_n = (F_{1,n}, ....,F_{d,n})$, $n\geq 1$, be a sequence of random vectors such that, for every $j=1,...,d$, the random variable $F_{j,n}$ belongs to a fixed Wiener chaos of a Gaussian field. We show that, as $n\to\infty$, the components of $F_n$ are asymptotically independent if and only if ${\rm Cov}(F_{i,n}^2,F_{j,n}^2)\to 0$ for every $i\neq j$. Our findings are based on a novel inequality for vectors of multiple Wiener-It\^o integrals, and represent a substantial refining of criteria for asymptotic independence in the sense of moments recently established by Nourdin and Rosinski.