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

Christian Döbler,Giovanni Peccati

doi:10.1214/18-ecp196

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 carre-du-champ operators. Our results substantially complete the fourth moment theorems recently obtained by Dobler and Peccati (2018) and Dobler, Vidotto and Zheng (2018). An important tool for achieving our goals is a novel product formula for multiple integrals under minimal conditions.

Highlights

The aim of the present paper is to provide an analytical study of FMTs for sequences of multiple stochastic integrals with respect to a general Poisson random measure
Our starting point will be the recent paper [DP18b], where we proved quantitative FMTs for multiple Wiener-Itô integrals with respect to a general Poisson random measure under mild regularity assumptions for the involved random variables
In order to be able to work in full generality, we will prove below a new general product formula that, under the minimal assumptions of the FMT of [DVZ18], ensures that certain linear combinations of contraction kernels are automatically in the corresponding L2 spaces

Summary

Introduction

Our starting point will be the recent paper [DP18b], where we proved quantitative FMTs for multiple Wiener-Itô integrals with respect to a general Poisson random measure under mild regularity assumptions for the involved random variables. Such regularity assumptions were later removed in [DVZ18], where the above mentioned fourth moment results were extended to the multivariate case. In order to be able to work in full generality, we will prove below a new general product formula that, under the minimal assumptions of the FMT of [DVZ18], ensures that certain linear combinations of contraction kernels are automatically in the corresponding L2 spaces. We use the symbol −D→ in order to indicate convergence in distribution of random variables

A new general product formula on the Poisson space

An extension of the fourth moment theorem

General Poisson point processes and technical framework

Proof of the product formula