Normal approximation and almost sure central limit theorem for non-symmetric Rademacher functionals

Abstract

In this work, we study the normal approximation and almost sure central limit theorems for some functionals of an independent sequence of Rademacher random variables. In particular, we provide a new chain rule that improves the one derived by Nourdin, Peccati and Reinert(2010) and then we deduce the bound on Wasserstein distance for normal approximation using the (discrete) Malliavin-Stein approach. Besides, we are able to give the almost sure central limit theorem for a sequence of random variables inside a fixed Rademacher chaos using the Ibragimov-Lifshits criterion.

Fourth moment theorems on the Poisson space in any dimension

“…As a consequence, one has as well as g ′ ∞ ≤ 2/π, g ′′ ∞ ≤ 2, see e.g. Section 2.3 in [44]. In what follows, we fix such a pair (f, g) of functions.…”

“…This yields a Peccati-Tudor type theorem, as well as an optimal improvement in the univariate case.Finally, a transfer principle "from-Poisson-to-Gaussian" is derived, which is closely related to the universality phenomenon for homogeneous multilinear sums.where d TV denotes the total variation distance between the laws of two real random variables. The techniques developed in [26] have also been adapted to non-Gaussian spaces which admit a Malliavin calculus structure: for instance, the papers [16,34,36,41] deal with the Poisson space case, whereas [17,18,30,44] develop the corresponding techniques for sequences of independent Rademacher random variables. The question FOURTH MOMENT THEOREMS 3 about general fourth moment theorems on these spaces, however, has remained open in general, until the two recent articles [13] and [11].…”

Almost sure central limit theorems in stochastic geometry

“…Therefore (16) follows since i (H n ) < ∞, for any i = 1, 2 and n ≥ 1, as a consequence of the following assumptions: Var(H n ) ∈ R + and I K,n ∈ R + for n ≥ 1, (29), (31), (25), and (26) (which hold for any n ≥ 1).…”

“…The main idea was to employ the Malliavin calculus on the Wiener space in order to check the conditions of the so-called Ibragimov-Lifshits criterion [7]. A similar approach was followed in [25] to provide an ASCLT for sequences of random variables belonging to a fixed Rademacher chaos.…”

A Peccati-Tudor type theorem for Rademacher chaoses

“…Concerning the normal approximation in [17] or [26], the authors were only able to obtain the bounds in some "smooth-version" distance, due to regularity involving in their chain rules and Stein's solution. In a follow-up work, Zheng [28] obtained a neater chain rule that requires minimal regularity (see [28,Remark 2.3]), from which he obtained the bound in Wasserstein distance as well as an almost sure central limit theorem for Rademacher chaos. It is worthy pointing out that without using any chain rule, the authors of [9,10] used carefully a representation of the discrete Malliavin gradient and the fundamental theorem of calculus to deduce the Berry-Esseen bound for normal approximation.…”