Convergence in Law Implies Convergence in Total Variation for Polynomials in Independent Gaussian, Gamma or Beta Random Variables

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 (1) the limit distribution is necessarily absolutely continuous with respect to the Lebesgue measure and (2) 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 Nourdin and Poly (Stoch Proc Appl 123:651–674, 2013) to the Gamma and Beta cases.