An invariance principle under the total variation distance

Abstract

Let X 1 , X 2 , . . . be a sequence of i.i.d. random variables, with mean zero and variance one. Let W n = (X 1 + . . . + X n )/ √ n. An old and celebrated result of Prohorov [16] asserts that W n converges in total variation to the standard Gaussian distribution if and only if W n 0 has an absolutely continuous component for some n 0 . In the present paper, we give yet another proof and extend Prohorov's theorem to a situation where, instead of W n , we consider more generally a sequence of homogoneous polynomials in the X i . More precisely, we exhibit conditions for a recent invariance principle proved by Mossel, O'Donnel and Oleszkiewicz [14] to hold under the total variation distance. There are many works about CLT under various metrics in the literature, but the present one seems to be the first attempt to deal with homogeneous polynomials in the X i with degree strictly greater than one.

Asymptotic development for the CLT in total variation distance

“…This is done especially in order to set up the main arguments and results from abstract Malliavin calculus coming from representation (2.3), that are used throughout this paper. Let us stress that Nourdin and Poly in [12] have dealt with r.v. 's fulfilling properties that imply (2.3), to which they apply results from [2] about a finite dimensional Malliavin type calculus.…”

“…Then, using a splitting procedure (see Proposition 3.1 for details), we may isolate this noise and achieve integration by parts formulae based on it. In the last years, a number of results concerning the weak convergence of functionals on the Wiener space using Malliavin calculus and Stein's method have been obtained by Nurdin, Peccati, Nualart and Poly; see, for example, [9,10,11,12]. In particular, in [9,10] the authors consider functionals living in a finite direct sum of chaoses and prove that under a very weak non-degeneracy condition (analogous to the one we consider here) the convergence in distribution of a sequence of such functionals implies the convergence in total variation.…”

Arbitrary order total variation Convergence of Markov semigroups using random grids

“…In [4], [5] and [9], it is used to study the Central Limit Theorem. Last but not least, in [7], the above splitting method (with 1 Br * (z * ,t) instead of φ δ (z − z * ,t √ n )) is used in a framework which is similar to the one in this paper.…”

Recent advances in various fields of numerical probability

“…This allows to derive integration by parts formulas and to obtain convenient estimates for the weights which appear in these formulas. It is worth mentioning that a variant of the Malliavin calculus based on a similar splitting method has already been used by Nourdin and Poly [59] (see also [60] and [54]). They use the so-called Γ-calculus (see e.g.…”