Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos

TL;DR

This paper establishes that homogeneous sums of independent variables exhibit universality in their Gaussian and chi-square approximations, allowing for simplified analysis via Wiener chaos techniques.

Contribution

It demonstrates that approximation bounds and convergence criteria for homogeneous sums can be transferred from Wiener chaos to general independent variables.

Findings

Explicit bounds for normal and chi-square approximations.

Universality of approximation properties across different distributions.

Extension of Wiener chaos bounds to homogeneous sums.

Abstract

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. 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.