Fourth moment theorems on the Poisson space: analytic statements via product formulae

TL;DR

This paper establishes necessary and sufficient conditions for the asymptotic normality of multiple integrals on the Poisson space, using novel product formulae and extending previous fourth moment theorems.

Contribution

It provides a comprehensive set of conditions for normal convergence of Poisson multiple integrals, enhancing the theoretical framework with new product formulae.

Findings

Complete characterization of asymptotic normality conditions.

Introduction of a new product formula for multiple integrals.

Extension of previous fourth moment theorems.

Abstract

We prove necessary and sufficient conditions for the asymptotic normality of multiple integrals with respect to a Poisson measure on a general measure space, expressed both in terms of norms of contraction kernels and of variances of carr\'e-du-champ operators. Our results substantially complete the fourth moment theorems recently obtained by D\"obler and Peccati (2018) and D\"obler, Vidotto and Zheng (2018). An important tool for achieving our goals is a novel product formula for multiple integrals under minimal conditions.