Let {F n } be a sequence of random variables belonging to a finite sum of Wiener chaoses. Assume further that it converges in distribution towards F ∞ satisfying Var(F ∞ ) > 0. Our first result is a sequential version of a theorem by Shigekawa [25]. More precisely, we prove, without additional assumptions, that the sequence {F n } actually converges in total variation and that the law of F ∞ is absolutely continuous. We give an application to discrete non-Gaussian chaoses. In a second part, we assume that each F n has more specifically the form of a multiple Wiener-Itô integral (of a fixed order) and that it converges in L 2 (Ω) towards F ∞ . We then give an upper bound for the distance in total variation between the laws of F n and F ∞ . As such, we recover an inequality due to Davydov and Martynova [6]; our rate is weaker compared to [6] (by a power of 1/2), but the advantage is that our proof is not only sketched as in [6]. Finally, in a third part we show that the convergence in the celebrated Peccati-Tudor theorem actually holds in the total variation topology.
We investigate the problem of finding necessary and sufficient conditions for convergence in distribution towards a general finite linear combination of independent chi-squared random variables, within the framework of random objects living on a fixed Gaussian space. Using a recent representation of cumulants in terms of the Malliavin calculus operators Γ i (introduced by Nourdin and Peccati in [13]), we provide conditions that apply to random variables living in a finite sum of Wiener chaoses. As an important by-product of our analysis, we shall derive a new proof and a new interpretation of a recent finding by Nourdin and Poly [16], concerning the limiting behaviour of random variables living in a Wiener chaos of order two. Our analysis contributes to a fertile line of research, that originates from questions raised by Marc Yor, in the framework of limit theorems for non-linear functionals of Brownian local times.
A. Inspired by the insightful article [4], we revisit the Nualart-Peccati-criterion [13] (now known as the Fourth Moment Theorem) from the point of view of spectral theory of general Markov diffusion generators. We are not only able to drastically simplify all of its previous proofs, but also to provide new settings of diffusive generators (Laguerre, Jacobi) where such a criterion holds. Convergence towards gamma and beta distributions under moment conditions is also discussed.
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.
The celebrated Nualart-Peccati criterion [Ann. Probab. 33 (2005) 177-193] ensures the convergence in distribution toward a standard Gaussian random variable N of a given sequence {Xn} n≥1 of multiple Wiener-Itô integrals of fixed order, if E[X
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