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2013 Absolute continuity and convergence of densities for random vectors on Wiener chaos
Ivan Nourdin, David Nualart, Guillaume Poly
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Electron. J. Probab. 18: 1-19 (2013). DOI: 10.1214/EJP.v18-2181


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.


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Ivan Nourdin. David Nualart. Guillaume Poly. "Absolute continuity and convergence of densities for random vectors on Wiener chaos." Electron. J. Probab. 18 1 - 19, 2013.


Accepted: 11 February 2013; Published: 2013
First available in Project Euclid: 4 June 2016

zbMATH: 1285.60053
MathSciNet: MR3035750
Digital Object Identifier: 10.1214/EJP.v18-2181

Primary: 60F05
Secondary: 60G15 , 60H05 , 60H07

Keywords: Convergence in distribution , Convergence in total variation , Malliavin calculus , multiple Wiener-Itô integral , Wiener Chaos

Vol.18 • 2013
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