Electronic Journal of Probability

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

Ivan Nourdin, David Nualart, and Guillaume Poly

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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.

Article information

Electron. J. Probab., Volume 18 (2013), paper no. 22, 19 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60G15: Gaussian processes 60H05: Stochastic integrals 60H07: Stochastic calculus of variations and the Malliavin calculus

Convergence in distribution Convergence in total variation Malliavin calculus multiple Wiener-Itô integral Wiener chaos

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Nourdin, Ivan; Nualart, David; Poly, Guillaume. Absolute continuity and convergence of densities for random vectors on Wiener chaos. Electron. J. Probab. 18 (2013), paper no. 22, 19 pp. doi:10.1214/EJP.v18-2181. https://projecteuclid.org/euclid.ejp/1465064247

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