# A bound on the 2-Wasserstein distance between linear combinations of independent random variables

## TL;DR

This paper establishes a bound on the 2-Wasserstein distance between linear combinations of independent random variables, aiding in quantifying convergence rates to second Wiener chaos elements using Malliavin-Stein techniques.

## Contribution

It introduces a new bound on the 2-Wasserstein distance applicable to sequences in the second Wiener chaos, extending Malliavin-Stein methods for quantitative convergence analysis.

## Key findings

- Bound effectively estimates Wasserstein distance for linear combinations of independent variables.
- Application to second Wiener chaos yields explicit convergence rates.
- Illustrative examples demonstrate the bound's versatility in various probabilistic settings.

## Abstract

We provide a bound on a natural distance between finitely and infinitely supported elements of the unit sphere of $\ell^2(\mathbb{N}^*)$, the space of real valued sequences with finite $\ell^2$ norm. We use this bound to estimate the 2-Wasserstein distance between random variables which can be represented as linear combinations of independent random variables. Our results are expressed in terms of a discrepancy measure which is related to Nourdin and Peccati's Malliavin-Stein method. The main area of application of our results is towards the computation of quantitative rates of convergence towards elements of the second Wiener chaos. After particularizing our bounds to this setting and comparing them with the available literature on the subject (particularly the Malliavin-Stein method for Variance-gamma random variables), we illustrate their versatility by tackling three examples: chi-squared approximation for second order $U$-statistics, asymptotics for sequences of quadratic forms and the behavior of the generalized Rosenblatt process at extreme critical exponent.