A four moments theorem for Gamma limits on a Poisson chaos

TL;DR

This paper establishes a four moments theorem for Gamma distribution limits on Poisson chaos, showing that convergence of third and fourth moments implies distributional convergence for specific chaos orders.

Contribution

It introduces a new four moments theorem for Gamma limits on Poisson chaos, with novel estimates for contraction norms for orders q=2 and q=4.

Findings

Convergence of third and fourth moments implies distributional convergence for q=2 and q=4.

Provides new estimates for contraction norms in Poisson chaos.

Applications to homogeneous sums and U-statistics on Poisson space.

Abstract

This paper deals with sequences of random variables belonging to a fixed chaos of order $q$ generated by a Poisson random measure on a Polish space. The problem is investigated whether convergence of the third and fourth moment of such a suitably normalized sequence to the third and fourth moment of a centred Gamma law implies convergence in distribution of the involved random variables. A positive answer is obtained for $q=2$ and $q=4$. The proof of this four moments theorem is based on a number of new estimates for contraction norms. Applications concern homogeneous sums and $U$-statistics on the Poisson space.