Empirical limit theorems for Wiener chaos

We consider a scheme of central limit theorem for empirical measures. In contrast to a classical empirical central limit theorem where the distribution governing the samples is fixed, we let the distribution change as the sample size grows in a triangular array setup. Certain asymptotic requirement is imposed on the changing distributions in order to create asymptotic uncorrelatedness of the empirical measures evaluated at disjoint subsets. This leads to an independently scattered Gaussian random measure as the limit. We establish weak convergence of multiple integrals with respect to the normalized empirical measures towards multiple Wiener-It\^o integrals. This empirical limit theorem is also extended to one involving an infinite series of multiple Wiener-It\^o integrals.