An Invariance Principle for Stochastic Series II. Non Gaussian Limits

Abstract

We study the convergence in total variation distance for series of the form $$ S_{N}(c,Z)=\sum_{l=1}^{N}\sum_{i_{1}<\cdots<i_{l}}c(i_{1},...,i_{l})Z_{i_{1}}\cdots Z_{i_{l}}, $$ where $Z_{k},k\in {\mathbb{N}}$ are independent centered random variables with ${\mathbb{E}}(Z_{k}^{2})=1$. This enters in the framework of the $U$--statistics theory which plays a major role in modern statistic. In the case when $Z_{k},k\in {\mathbb{N}}$ are standard normal, $S_{N}(c,Z)$ is an element of the sum of the first $N$ Wiener chaoses and, starting with the seminal paper of D. Nualart and G. Peccati, the convergence of such functionals to the Gaussian law has been extensively studied. So the interesting point consists in studying invariance principles, that is, to replace Gaussian random variables with random variables with a general law. This has been done in several papers using the Fortet--Mourier distance, the Kolmogorov distance or the total variance distance. In particular, estimates of the total variance distance in terms of the fourth order cumulants has been given in the part I (Gaussian limits) of the present paper. But, as the celebrated Fourth Moment Theorem of Nualart and Peccati shows, such estimates are pertinent to deal with Gaussian limits. In the present paper we study the convergence to general limits which may be non Gaussian, and then the estimates of the error has to be done in terms of the low influence factor only.