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2017 On the sub-Gaussianity of the Beta and Dirichlet distributions
Olivier Marchal, Julyan Arbel
Electron. Commun. Probab. 22: 1-14 (2017). DOI: 10.1214/17-ECP92

Abstract

We obtain the optimal proxy variance for the sub-Gaussianity of Beta distribution, thus proving upper bounds recently conjectured 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 derive the optimal proxy variance for the Dirichlet distribution, which is apparently a novel result. We also provide a new proof of the optimal proxy variance for the Bernoulli distribution, and discuss in this context the proxy variance relation to log-Sobolev inequalities and transport inequalities.

Citation

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Olivier Marchal. Julyan Arbel. "On the sub-Gaussianity of the Beta and Dirichlet distributions." Electron. Commun. Probab. 22 1 - 14, 2017. https://doi.org/10.1214/17-ECP92

Information

Received: 19 June 2017; Accepted: 4 October 2017; Published: 2017
First available in Project Euclid: 13 October 2017

zbMATH: 06797807
MathSciNet: MR3718704
Digital Object Identifier: 10.1214/17-ECP92

Subjects:
Primary: 97K50

JOURNAL ARTICLE
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