Central Limit Theorems for Sequences of Multiple Stochastic

We characterize the convergence in distribution to a standard normal law for a sequence of multiple stochastic integrals of a fixed order with variance converging to 1. Some applications are given, in particular to study the limiting behavior of quadratic functionals of Gaussian processes. 1. Introduction. In this paper, we characterize the convergence in distribution to a normal N (0, 1) law for a sequence of random variables F k belonging to a fixed Wiener chaos, and with variance tending to 1. We show that a necessary and sufficient condition for this convergence is that the moment of fourth order of F k converges to 3. If F k is a multiple stochas-tic integral of order n of a symmetric and square integrable kernel f k , for instance on [0, 1] n , another necessary and sufficient condition for the above convergence is that, for all p = 1,. .. , n − 1, the contractions of order p (defined by f