An invariance principle under the total variation distance

TL;DR

This paper extends an invariance principle under total variation distance to homogeneous polynomials of degree greater than one, providing conditions for convergence to Gaussian distributions in a more general setting.

Contribution

It generalizes Prohorov's invariance principle to homogeneous polynomials of degree greater than one under total variation distance.

Findings

Provides new conditions for invariance principles involving homogeneous polynomials

Extends total variation convergence results beyond linear sums

First to address CLT under total variation for higher-degree polynomials

Abstract

Let $X_1,X_2,\ldots$ be a sequence of i.i.d. random variables, with mean zero and variance one. Let $W_n=(X_1+\ldots+X_n)/\sqrt{n}$. An old and celebrated result of Prohorov asserts that $W_n$ converges in total variation to the standard Gaussian distribution if and only if $W_{n_0}$ has an absolutely continuous component for some $n_0$. In the present paper, we give yet another proof and extend Prohorov's theorem to a situation where, instead of $W_n$, we consider more generally a sequence of homogoneous polynomials in the $X_i$. More precisely, we exhibit conditions for a recent invariance principle proved by Mossel, O'Donnel and Oleszkiewicz to hold under the total variation distance. There are many works about CLT under various metrics in the literature, but the present one seems to be the first attempt to deal with homogeneous polynomials in the $X_i$ with degree strictly greater than one.