Strong asymptotic independence on Wiener chaos

Abstract

Let $F_n=(F_{1,n},....,F_{d,n})$, $n\ge 1$, 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 $\mathrm{Cov}(F_{i,n}^2,F_{j,n}^2)\to 0$ for every $i\ne j$. Our findings are based on a novel inequality for vectors of multiple Wiener-Itô integrals, and represent a substantial refining of criteria for asymptotic independence in the sense of moments recently established by Nourdin and Rosinski.

13 pages

References in corpus (1)

• Central limit theorems for sequences of multiple stochastic integrals

Cited by in corpus (1)

• Malliavin-Stein method for Variance-Gamma approximation on Wiener space