Nourdin-Peccati analysis on Wiener and Wiener-Poisson space for general distributions

TL;DR

This paper extends Nourdin-Peccati analysis to general distributions on Wiener and Wiener-Poisson spaces, providing bounds, convergence conditions, and applications to Gaussian functionals and fractional Brownian motion.

Contribution

It generalizes Nourdin-Peccati techniques to broader distributions and spaces, offering new bounds, convergence criteria, and applications to non-central limit theorems.

Findings

Universal bounds on derivatives of Stein solutions

Conditions for convergence to Normal in Wiener and Wiener-Poisson spaces

Application to fractional Brownian motion with Hurst > 1/2

Abstract

Given a reference random variable, we study the solution of its Stein equation and obtain universal bounds on its first and second derivatives. We then extend the analysis of Nourdin and Peccati by bounding the Fortet-Mourier and Wasserstein distances from more general random variables such as members of the Exponential and Pearson families. Using these results, we obtain non-central limit theorems, generalizing the ideas applied to their analysis of convergence to Normal random variables. We do these in both Wiener space and the more general Wiener-Poisson space. In the former, we study conditions for convergence under several particular cases and characterize when two random variables have the same distribution. As an example, we apply this tool to bilinear functionals of Gaussian subordinated fields where the underlying process is a fractional Brownian motion with Hurst parameter bigger than 1/2. In the latter space we give suficient conditions for a sequence of multiple (Wiener-Poisson) integrals to converge to a Normal random variable.