The fourth moment theorem on the Poisson space

Abstract

We prove a fourth moment bound without remainder for the normal approximation of random variables belonging to the Wiener chaos of a general Poisson random measure. Such a result-that has been elusive for several years-shows that the so-called 'fourth moment phenomenon', first discovered by Nualart and Peccati [Ann. Probab. 33 (2005) 177-193] in the context of Gaussian fields, also systematically emerges in a Poisson framework. Our main findings are based on Stein's method, Malliavin calculus and Mecketype formulae, as well as on a methodological breakthrough, consisting in the use of carré-du-champ operators on the Poisson space for controlling residual terms associated with add-one cost operators. Our approach can be regarded as a successful application of Markov generator techniques to probabilistic approximations in a nondiffusive framework: as such, it represents a significant extension of the seminal contributions by Ledoux [Ann. Probab. 40 (2012) 2439-2459 and Azmoodeh, Campese and Poly [J. Funct. Anal. 266 (2014) 2341-2359]. To demonstrate the flexibility of our results, we also provide some novel bounds for the Gamma approximation of nonlinear functionals of a Poisson measure.

Fourth moment theorems on the Poisson space in any dimension

“…Furthermore, still keeping the spectral point of view as in [13], by replacing the rather intrinsic techniques used there with an adaption of a recent construction of exchangeable pairs couplings from [32], we can even remove certain technical conditions which seem inevitable in order to justify the computations in [13]. In this way, we are able to prove our results under the weakest possible assumption of finite fourth moments.…”

“…Outline. In the recent paper [13], the authors succeeded in proving exact quantitative fourth moment theorems for multiple Wiener-Itô integrals on the Poisson space. Briefly, their method consisted in extending the spectral framework initiated by the remarkable paper [22], and further refined by [1], from the situation of a diffusive Markov generator to the non-diffusive Ornstein-Uhlenbeck generator on the Poisson space.…”

“…Briefly, their method consisted in extending the spectral framework initiated by the remarkable paper [22], and further refined by [1], from the situation of a diffusive Markov generator to the non-diffusive Ornstein-Uhlenbeck generator on the Poisson space. The principal aim of the present article is to extend the results from [13] to the multivariate case of vectors of multiple integrals. In view of the result of Peccati and Tudor [35] on vectors of multiple integrals on a Gaussian space, we are in particular interested in discussing the relationship between coordinatewise convergence and joint convergence to normality.…”

“…In this way, we are able to prove our results under the weakest possible assumption of finite fourth moments. In the univariate case, our strategy provides an optimal improvement of the Wasserstein bound given in Theorem 1.3 of [13] and, a fortiori, of the associated qualitative fourth moment theorem on the Poisson space (see Corollary 1.4 in [13]).…”

“…Fix an integer q ≥ 1 and let F ∈ C q be such that σ 2 := E[F 2 ] > 0 and E[F 4 ] < ∞. Then, with N denoting a standard normal random variable, we have the bounds: [13], respectively, since they do not require any additional regularity from the involved multiple integrals like e.g. Assumption A in [13].…”