Convergence in total variation on Wiener chaos

Abstract

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.

Convergence in Law Implies Convergence in Total Variation for Polynomials in Independent Gaussian, Gamma or Beta Random Variables

“…More precisely, we shall prove the following result, which may be seen as an extension to the Gamma and Beta cases of our previous results in [9]. Theorem 1.1 Assume that one of the following three conditions is satisfied:…”

Asymptotic Cramér Type Decomposition for Wiener and Wigner Integrals

“…We are thus interested in asymptotic decomposition (1) and we prove (for multiple Wiener integrals) a stronger result than the one contained in [Tu11]. The proof we propose below in this context is new, short and independent from Cramér's original result: it is based on a recent result by Nourdin and Poly on convergence in total variation on Wiener chaoses (see [NP12]) as well as on the fact that on the Wiener chaoses, the central convergence is controlled by the convergence of the moments of order 2 and 4 (see [NO08,NP05]). Such a phenomenon has been recently furtherly investigated in [BBN + 11] where optimal Berry-Esséen rates are given in terms of third and fourth cumulants.…”

“…• If σ 1 > 0 and σ 2 > 0 then the convergence to N 2 0, diag σ 2 1 , σ 2 2 holds in total variation, see Theorem 5.2 in [NP12].…”

“…Since X n + Y n is tight, Lemma 2.4 in [NP12] implies that X n + Y n is uniformly bounded in all the L p . But as E [X n Y n ] = 0, both X n and Y n are uniformly bounded in L 2 , and in all the L p as well by hypercontractivity.…”

Regularity of Wiener functionals under a Hörmander type condition of order one

“…At our knowledge there are not many results concerning general vectors on the Wiener space -except of course the celebrated criterion given by Malliavin and the Bouleau Hirsh criterion for the absolute continuity. Another criterion proved by Kusuoka in [6] and further generalized by Nourdin and Poly [11] and Nualart, Nourdin and Poly [12] concerns vectors living in a finite number of chaoses. All these criterions suppose that the determinant of the Malliavin covariance matrix is non null in a more or less strong sense -but give no hint about the possible analysis of this condition.…”