Stable limit theorems on the Poisson space

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 Wasserstein 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 G. Peccati, J. L. Solé, M. S. Taqqu & F. Utzet. “Stein’s method and normal approximation of Poisson functionals” for Gaussian approximations; and by G. Peccati “The Chen-Stein method for Poisson functionals” for Poisson approximations; as well as the recently established fourth-moment theorem on the Poisson space of C. Döbler & G.Peccati “The fourth moment theorem on the Poisson space”. We give an application to stochastic processes.