An invariance principle for stochastic series I. Gaussian limits

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.