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. Sole, 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. Dobler & G.Peccati “The fourth moment theorem on the Poisson space”. We give an application to stochastic processes.
Highlights
One of the celebrated contributions of Rényi [36, 37] is a refinement of the notion of convergence in law, commonly referred to as stable convergence
Stable convergence is tailored for studying conditional limits of sequences of random variables
A stable limit is, typically, a mixture, that is, in our terminology: a random variable whose law depends on a random parameter; for instance, a centered Gaussian random variable with random variance, or a Poisson random variable with random mean
Summary
One of the celebrated contributions of Rényi [36, 37] is a refinement of the notion of convergence in law, commonly referred to as stable convergence. The crucial tool to establish our results is a duality relation ( referred to as integration by parts) between the operators D and δ: EF δu = Eν(uDF ) This relation is at the heart of the Malliavin-Stein approach to obtain limit theorems both in a Gaussian [24, Chapter 5] and in a Poisson setting [28].
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