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Absolute continuity and convergence of densities for random vectors on Wiener chaos. (English) Zbl 1285.60053
Authors’ 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 S. Kusuoka [J. Fac. Sci., Univ. Tokyo, Sect. I A 30, 191–197 (1983; Zbl 0523.60039)]. 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 I. Nourdin and G. Poly [Stochastic Processes Appl. 123, No. 2, 651–674 (2013; Zbl 1259.60029)]. This follows from an inequality relating the total variation distance with the Fortet-Mourier distance. Finally, applications to some particular cases are discussed.

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