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Absolute continuity and convergence of densities for random vectors on Wiener chaos
Nourdin, Ivan Nualart, David Poly, Guillaume
Nourdin, Ivan
Nualart, David
Poly, Guillaume
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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.
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This is the publisher's version, also available electronically from http://ejp.ejpecp.org/article/view/2181
Date
2013-02-11
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Institute of Mathematical Statistics (IMS)
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Citation
Nourdin et al. (2013). Absolute continuity and convergence of densities for random vectors on Wiener chaos. Electronic Journal of Probability 18:41658.