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

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1.	Title	Title of document	Absolute continuity and convergence of densities for random vectors on Wiener chaos
2.	Creator	Author's name, affiliation, country	Ivan Nourdin; Université de Lorraine; France
2.	Creator	Author's name, affiliation, country	David Nualart; University of Kansas; United States
2.	Creator	Author's name, affiliation, country	Guillaume Poly; Luxembourg University; Luxembourg
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Convergence in distribution; Convergence in total variation; Malliavin calculus; multiple Wiener-Itô integral; Wiener chaos
3.	Subject	Subject classification	60F05; 60G15; 60H05; 60H07
4.	Description	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.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)	<ul><li>Bally, Vlad; Caramellino, Lucia. Riesz transform and integration by parts formulas for random variables. <EM>Stochastic Process. Appl.</EM> 121 (2011), no. 6, 1332--1355. <a href="http:ANR-09-BLAN-0114, ANR-10-BLAN-0121, NSF grant DMS-1208625
7.	Date	(YYYY-MM-DD)	2013-02-11
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/2181
10.	Identifier	Digital Object Identifier	10.1214/EJP.v18-2181
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 18
12.	Language	English=en	en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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