Fourth moment theorems on the Poisson space in any dimension

Abstract

We extend to any dimension the quantitative fourth moment theorem on the Poisson setting, recently proved by C. . In particular, by adapting the exchangeable pairs couplings construction introduced by I. Nourdin and G. Zheng (2017) to the Poisson framework, we prove our results under the weakest possible assumption of finite fourth moments. 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].

A Peccati-Tudor type theorem for Rademacher chaoses

“…Similar universality result for Poisson chaos was first established in [24] and refined recently in [7]. It was pointed out in [24] and [18] that homogeneous sums inside the Rademacher chaos are not universal with respect to normal approximation and a counterexample is available e.g.…”

“…Recall first that two random variables W and W , defined on a common probability space, are said to form an exchangeable pair, if (W, W ) has the same distribution as (W , W). Proposition 1.1 (Proposition 3.5 in [7]). For each t > 0, let (F, F t ) be an exchangeable pair of centred d-dimensional random vectors defined on a common probability space.…”

“…For the proof, one can refer to [7,Proposition 3.3]. One may also want to refer to Theorem 3.5 of [17] for a different coupling bound.…”

Malliavin–Stein method: a survey of some recent developments

“…Such a Markov triple is associated with the space of square-integrable functionals of a Poisson measure on a general pair (Z, Z ), where Z is a Polish space and Z is the associated Borel σ -field. The requirement that Z is Polish -together with several other assumptions adopted in the present section -is made in order to simplify the discussion; the reader is referred to [37,38] for statements and proofs in the most general setting. See also [50,51] for an exhaustive presentation of tools of stochastic analysis for functionals of Poisson processes, as well as [81] for a discussion of the relevance of variational techniques in the framework of modern stochastic geometry.…”

“…On the one hand, we aim at presenting the essence of the Malliavin-Stein method for functionals of Gaussian fields, by discussing the crucial elements of Malliavin calculus and Stein's method together with their interaction (see Section 2 and Section 3). On the other hand, we aim at introducing the reader to some of the most recent developments of the theory, with specific focus on the general theory of Markov semigroups in a diffusive setting (following the seminal references [52,5], as well as [73,53,54]), and on integration by parts formulae (and associated operators) in the context of functionals of a random point measure [37,38,55,49,48,90]. This corresponds to the content of Section 4 and Section 5, respectively.…”