Convergence in law implies convergence in total variation for polynomials in independent Gaussian, Gamma or Beta random variables

TL;DR

This paper proves that for polynomials of bounded degree evaluated in independent Gaussian, Gamma, or Beta variables, convergence in law to a nonconstant distribution guarantees convergence in total variation, with the limit being absolutely continuous.

Contribution

It extends previous results to Gamma and Beta distributions, showing that law convergence implies total variation convergence for such polynomial sequences.

Findings

Limit distributions are necessarily absolutely continuous.

Convergence in law implies convergence in total variation.

Results extend to Gamma and Beta distributions.

Abstract

Consider a sequence of polynomials of bounded degree evaluated in independent Gaussian, Gamma or Beta random variables. We show that, if this sequence converges in law to a nonconstant distribution, then (i) the limit distribution is necessarily absolutely continuous with respect to the Lebesgue measure and (ii) the convergence automatically takes place in the total variation topology. Our proof, which relies on the Carbery-Wright inequality and makes use of a diffusive Markov operator approach, extends the results of \cite{NP} to the Gamma and Beta cases.