Fourth moment theorems on the Poisson space in any dimension

TL;DR

This paper generalizes the fourth moment theorem on the Poisson space to any dimension, using exchangeable pairs, and introduces a transfer principle linking Poisson and Gaussian universality phenomena.

Contribution

It extends the fourth moment theorem to higher dimensions on the Poisson space with minimal assumptions and develops a transfer principle connecting Poisson and Gaussian limits.

Findings

Proves a multidimensional fourth moment theorem on Poisson space.

Establishes a transfer principle from Poisson to Gaussian limits.

Provides an optimal univariate case improvement.

Abstract

We extend to any dimension the quantitative fourth moment theorem on the Poisson setting, recently proved by C. D\"obler and G. Peccati (2017). 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.