# On the sub-Gaussianity of the Beta and Dirichlet distributions

## TL;DR

This paper determines the optimal sub-Gaussian proxy variance for Beta and Dirichlet distributions, providing new proofs and extending results to these distributions, with implications for inequalities in probability theory.

## Contribution

It establishes the first known optimal proxy variance for the Dirichlet distribution and offers novel proof techniques for Beta distribution sub-Gaussianity.

## Key findings

- Optimal proxy variance for Beta distribution proved
- Derived the first proxy variance for Dirichlet distribution
- Connected proxy variance to log-Sobolev and transport inequalities

## 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.