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) = ∞ m=1 α1<...<αm c(α 1 , ..., α m ) m i=1 Z αi where Z k , k ∈ 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.