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

Ivan Nourdin (1) , Guillaume Poly (2)

1 IECL - Institut Élie Cartan de Lorraine
2 FSTC - Faculté des Sciences, de la Technologie et de la Communication

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.

Mots clés: log-concave distribution; orthogonal polynomials; Convergence in law; absolute continuity; convergence in total variation; Carbery-Wright inequality

Domaines: Probabilités [math.PR]

Ivan Nourdin, Guillaume Poly. Convergence in law implies convergence in total variation for polynomials in independent Gaussian, Gamma or Beta random variables. 2013. ⟨hal-00821911⟩