On the sub-Gaussianity of the Beta and Dirichlet distributions

Olivier Marchal (1) , Julyan Arbel (2)

1 PSPM - Probabilités, statistique, physique mathématique
2 MISTIS - Modelling and Inference of Complex and Structured Stochastic Systems

We obtain the optimal proxy variance for the sub-Gaussianity of Beta distributions, thus proving, and improving, a recent conjecture made by Elder (2016). We provide different proof techniques for the symmetrical (around its mean) case and the non-symmetrical case. The technique in the latter case relies on studying the ordinary differential equation satisfied by the Beta moment-generating function known as the confluent hypergeometric function. As a consequence, we also derive the optimal proxy variance for Dirichlet distributions.

Domaines: Probabilités [math.PR]; Statistiques [math.ST]

Olivier Marchal, Julyan Arbel. On the sub-Gaussianity of the Beta and Dirichlet distributions. Electronic Communications in Probability, 2017, 22 (paper no. 54), pp.1-14. ⟨10.1214/17-ECP92⟩. ⟨hal-01521300⟩