Stein's method on the second Wiener chaos : 2-Wasserstein distance

TL;DR

This paper introduces a novel Fourier-based Stein characterization for the second Wiener chaos, providing new bounds on the 2-Wasserstein distance without relying on traditional Stein equation solutions.

Contribution

It presents an original proof technique connecting Fourier analysis and Stein's method for the second Wiener chaos, with new discrepancy measures and applications.

Findings

Derived Stein characterizations for second Wiener chaos variables

Established bounds on 2-Wasserstein distance using the new discrepancy

Compared results with existing methods, demonstrating advantages

Abstract

In the first part of the paper we use a new Fourier technique to obtain a Stein characterizations for random variables in the second Wiener chaos. We provide the connection between this result and similar conclusions that can be derived using Malliavin calculus. We also introduce a new form of discrepancy which we use, in the second part of the paper, to provide bounds on the 2-Wasserstein distance between linear combinations of independent centered random variables. Our method of proof is entirely original. In particular it does not rely on estimation of bounds on solutions of the so-called Stein equations at the heart of Stein's method. We provide several applications, and discuss comparison with recent similar results on the same topic.