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On the sub-Gaussianity of the Beta and Dirichlet distributions

Olivier Marchal 1 Julyan Arbel 2
1 PSPM - Probabilités, statistique, physique mathématique
ICJ - Institut Camille Jordan [Villeurbanne]
2 MISTIS - Modelling and Inference of Complex and Structured Stochastic Systems
Inria Grenoble - Rhône-Alpes, Grenoble INP - Institut polytechnique de Grenoble - Grenoble Institute of Technology, LJK - Laboratoire Jean Kuntzmann
Abstract : 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.
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Submitted on : Friday, January 26, 2018 - 11:54:56 AM
Last modification on : Wednesday, November 3, 2021 - 5:59:23 AM

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

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