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arXiv · 1705.00048

On the sub-Gaussianity of the Beta and Dirichlet distributions

Olivier Marchal·Julyan Arbel
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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.

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Olivier Marchal, Julyan Arbel. 2017-09-25. On the sub-Gaussianity of the Beta and Dirichlet distributions. https://doi.org/10.1214/17-ecp92

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