## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 22 (2017), paper no. 54, 14 pp.

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

Olivier Marchal and Julyan Arbel

#### 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.

#### Article information

**Source**

Electron. Commun. Probab. Volume 22 (2017), paper no. 54, 14 pp.

**Dates**

Received: 19 June 2017

Accepted: 4 October 2017

First available in Project Euclid: 13 October 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1507860211

**Digital Object Identifier**

doi:10.1214/17-ECP92

**Subjects**

Primary: 97K50: Probability theory

**Keywords**

sub-Gaussian Beta distribution Dirichlet distribution concentration inequality transport inequality log-Sobolev inequality

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Marchal, Olivier; Arbel, Julyan. On the sub-Gaussianity of the Beta and Dirichlet distributions. Electron. Commun. Probab. 22 (2017), paper no. 54, 14 pp. doi:10.1214/17-ECP92. https://projecteuclid.org/euclid.ecp/1507860211