Invariance Principles for Homogeneous Sums: Universality of Gaussian Wiener Chaos

We compute explicit bounds in the normal and chi-square approximations of multilinear homogenous sums (of arbitrary order) of general centered independent random variables with unit variance. Our techniques combine an invariance principle by Mossel, O’Donnell and Oleszkiewicz with a refinement of some recent results by Nourdin and Peccati, about the approximation of laws of random variables belonging to a fixed (Gaussian) Wiener chaos. In particular, we show that chaotic random variables enjoy the following form of universality : (a) the normal and chi-square approximations of any homogenous sum can be completely characterized and assessed by first switching to its Wiener chaos counterpart, and (b) the simple upper bounds and convergence criteria available on the Wiener chaos extend almost verbatim to the class of homogeneous sums. These results partially rely on the notion of “low influences” for functions defined on product spaces, and provide a generalization of central and non-central limit theorems proved by Nourdin, Nualart and Peccati. They also imply a further drastic simplification of the method of moments and cumulants – as applied to the proof of probabilistic limit theorems – and yield substantial generalizations, new proofs and new insights into some classic findings by de Jong and Rotar’. Our tools involve the use of Malliavin calculus, and of both the Stein’s method and the Lindeberg invariance principle for probabilistic approximations.