On the second order Poincar\'e inequality and CLT on Wiener-Poisson space

TL;DR

This paper develops a second order Poincaré inequality on Wiener-Poisson space, providing bounds for Wasserstein distance and CLT applications for various stochastic functionals, including Gaussian and Poisson processes.

Contribution

It introduces a new second order Poincaré inequality on Wiener-Poisson space and demonstrates its utility in deriving CLT bounds for complex stochastic functionals.

Findings

Derived a Wasserstein distance bound in Wiener-Poisson space.

Established a second order Poincaré inequality useful for computations.

Applied results to CLTs for Gaussian and Poisson functionals.

Abstract

An upper bound for the Wasserstein distance is provided in the general framework of the Wiener-Poisson space. Is obtained from this bound a second order Poincar\'e-type inequality which is useful in terms of computations. For completeness sake, is made a survey of these results on the Wiener space, the Poisson space, and the Wiener-Poisson space, and showed several applications to central limit theorems with relevant examples: linear functionals of Gaussian subordinated field (where the subordinated field can be processes like fractional Brownian motion or the solution of the Ornstein-Uhlenbeck SDE driven by fractional Brownian motion), Poisson functionals in the first Poisson chaos restricted to \small" jumps (particularly fractional L\'evy processes) and the product of two Ornstein-Uhlenbeck processes (one in the Wiener space and the other in the Poisson space). Also, are obtained bounds for their rate of convergence to normality.