Absolute continuity and convergence of densities for random vectors on Wiener chaos

TL;DR

This paper investigates the conditions under which vectors in Wiener chaos are absolutely continuous and their densities converge, establishing new criteria and bounds, and extending known results to multivariate cases.

Contribution

It provides new criteria for absolute continuity, bounds on annihilating polynomials, and extends convergence results from law to total variation for Wiener chaos vectors.

Findings

Probability of vanishing Malliavin matrix determinant is 0 or 1.

Convergence in law implies convergence in total variation.

Extended multivariate convergence results to Wiener chaos vectors.

Abstract

The aim of this paper is to establish some new results on the absolute continuity and the convergence in total variation for a sequence of d-dimensional vectors whose components belong to a finite sum of Wiener chaoses. First we show that the probability that the determinant of the Malliavin matrix of such vectors vanishes is zero or one, and this probability equals to one is equivalent to say that the vector takes values in the set of zeros of a polynomial. We provide a bound for the degree of this annihilating polynomial improving a result by Kusuoka. On the other hand, we show that the convergence in law implies the convergence in total variation, extending to the multivariate case a recent result by Nourdin and Poly. This follows from an inequality relating the total variation distance with the Fortet-Mourier distance. Finally, applications to some particular cases are discussed.