Central limit theorems for double Poisson integrals

TL;DR

This paper establishes central limit theorems for double Poisson integrals, providing conditions for convergence to Gaussian distributions based on kernel contractions, with applications to Ornstein-Uhlenbeck processes and hazard rates.

Contribution

It introduces new kernel contraction conditions for Gaussian convergence of double Poisson integrals, avoiding reliance on asymptotic dependence properties.

Findings

Conditions for Gaussian convergence are expressed via kernel contractions.

The approach simplifies analysis by avoiding mixing or dependence assumptions.

Applications include functionals of Ornstein-Uhlenbeck processes and hazard rates.

Abstract

Motivated by second order asymptotic results, we characterize the convergence in law of double integrals, with respect to Poisson random measures, toward a standard Gaussian distribution. Our conditions are expressed in terms of contractions of the kernels. To prove our main results, we use the theory of stable convergence of generalized stochastic integrals developed by Peccati and Taqqu. One of the advantages of our approach is that the conditions are expressed directly in terms of the kernel appearing in the multiple integral and do not make any explicit use of asymptotic dependence properties such as mixing. We illustrate our techniques by an application involving linear and quadratic functionals of generalized Ornstein--Uhlenbeck processes, as well as examples concerning random hazard rates.