Stable limit theorems on the Poisson space

TL;DR

This paper establishes stable limit theorems for functionals of Poisson point processes using Malliavin calculus, covering Gaussian and Poisson limits with quantitative bounds and applications to stochastic processes.

Contribution

It generalizes existing limit theorems on the Poisson space by providing new stable convergence results with explicit conditions and bounds, extending prior Gaussian and Poisson approximation results.

Findings

Proves stable limit theorems for Poisson functionals.

Provides quantitative bounds in the Gaussian case.

Extends classical limit theorems with new conditions and applications.

Abstract

We prove limit theorems for functionals of a Poisson point process using the Malliavin calculus on the Poisson space. The target distribution is conditionally either a Gaussian vector or a Poisson random variable. The convergence is stable and our conditions are expressed in terms of the Malliavin operators. For conditionally Gaussian limits, we also obtain quantitative bounds, given for the Monge-Kantorovich transport distance in the univariate case; and for another probabilistic variational distance in higher dimension. Our work generalizes several limit theorems on the Poisson space, including the seminal works by Peccati, Sol\'e, Taqqu & Utzet for Gaussian approximations; and by Peccati for Poisson approximations; as well as the recently established fourth-moment theorem on the Poisson space of D\"obler & Peccati. We give an application to stochastic processes.