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 [8]. 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 [11]. This follows from an inequality relating the total variation distance with the Fortet-Mourier distance. Finally, applications to some particular cases are discussed.
Highlights
The purpose of this paper is to establish some new results on the absolute continuity and the convergence of the densities in some Lp(Rd) for a sequence of d-dimensional random vectors whose components belong to finite sum of Wiener chaos
Given two d-dimensional random vectors F and G, we denote by dT V (F, G) the total variation distance between the laws of F and G, defined by dT V (F, G) = sup |P (F ∈ A) − P (G ∈ A)|, A∈B(Rd) where the supremum is taken over all Borel sets A of Rd
There is an equivalent formulation for dT V, which is often useful: dT V (F, G)
Summary
The purpose of this paper is to establish some new results on the absolute continuity and the convergence of the densities in some Lp(Rd) for a sequence of d-dimensional random vectors whose components belong to finite sum of Wiener chaos. These result generalize previous works by Kusuoka [8] and by Nourdin and Poly [11], and are based on a combination of the techniques of Malliavin calculus, the Carbery-Wright inequality and some recent work on algebraic dependence for a family of polynomials.
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