Universal Gaussian fluctuations on the discrete Poisson chaos

TL;DR

This paper establishes that homogeneous sums within a fixed discrete Poisson chaos exhibit universal Gaussian fluctuations, extending known results from Gaussian settings and refining existing CLTs for Poisson space variables.

Contribution

It demonstrates universality of Gaussian fluctuations in discrete Poisson chaos and refines central limit theorems for Poisson functionals.

Findings

Homogeneous sums in Poisson chaos are universal for normal approximations.

Refinements of CLTs for Poisson space variables are provided.

Results parallel Gaussian chaos findings, extending their applicability.

Abstract

We prove that homogenous sums inside a fixed discrete Poisson chaos are universal with respect to normal approximations. This result parallels some recent findings, in a Gaussian context, by Nourdin, Peccati and Reinert (2010). As a by-product of our analysis, we provide some refinements of the CLTs for random variables on the Poisson space proved by Peccati, Sol\'e, Taqqu and Utzet (2010), and by Peccati and Zheng (2010).