  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.
