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

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Issue Date
2013-02-11Author
Nourdin, Ivan
Nualart, David
Poly, Guillaume
Publisher
Institute of Mathematical Statistics (IMS)
Type
Article
Article Version
Scholarly/refereed, publisher version
Metadata
Show full item recordAbstract
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.
Description
This is the publisher's version, also available electronically from http://ejp.ejpecp.org/article/view/2181
ISSN
1083-6489Collections
Citation
Nourdin et al. (2013). Absolute continuity and convergence of densities for random vectors on Wiener chaos. Electronic Journal of Probability 18:41658.
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