An invariance principle for stochastic series I. Gaussian limits

TL;DR

This paper establishes an invariance principle for stochastic series, demonstrating convergence to Gaussian limits for sequences of independent variables with general distributions, and provides error estimates in total variation distance.

Contribution

It extends invariance principles to non-Gaussian variables in stochastic series and offers quantitative error bounds in total variation distance.

Findings

Proves convergence to Gaussian limits for general independent variables.

Provides explicit error estimates in total variation distance.

Extends classical results from Wiener chaos to broader settings.

Abstract

We study invariance principles and convergence to a Gaussian limit for stochastic series of the form $S(c,Z)=\sum_{m=1}^{\infty }\sum_{\alpha _{1}<...<\alpha _{m}}c(\alpha _{1},...,\alpha _{m})\prod_{i=1}^{m}Z_{\alpha _{i}}$ where $Z_{k}$, $k\in \mathbb{N}$, is a sequence of centred independent random variables of unit variance. In the case when the $Z_{k}$'s are Gaussian, $S(c,Z)$ is an element of the Wiener chaos and convergence to a Gaussian limit (so the corresponding nonlinear CLT) has been intensively studied by Nualart, Peccati, Nourdin and several other authors. The invariance principle consists in taking $Z_{k}$ with a general law. It has also been considered in the literature, starting from the seminal papers of Jong, and a variety of applications including $U$-statistics are of interest. Our main contribution is to study the convergence in total variation distance and to give estimates of the error.