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Consider a sequence of polynomials of bounded degree evaluated in independentGaussian, Gamma or Beta random variables. We show that, if this sequenceconverges in law to a nonconstant distribution, then (i) the limit distributionis 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 adiffusive Markov operator approach, extends the results of \cite{NP} to theGamma and Beta cases.

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