On strict sub-Gaussianity, optimal proxy variance and symmetry for bounded random variables

¹ Université Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, 38000 Grenoble, France.
² Université de Lyon, CNRS UMR 5208, Université Jean Monnet, Institut Camille Jordan, 69000 Lyon, France.
³ Department of Mathematics and Statistics, La Trobe University, Bundoora Melbourne 3086, Victoria, Australia.

^* Corresponding author: julyan.arbel@inria.fr

Received: 18 April 2019
Accepted: 13 July 2019

Abstract

We investigate the sub-Gaussian property for almost surely bounded random variables. If sub-Gaussianity per se is de facto ensured by the bounded support of said random variables, then exciting research avenues remain open. Among these questions is how to characterize the optimal sub-Gaussian proxy variance? Another question is how to characterize strict sub-Gaussianity, defined by a proxy variance equal to the (standard) variance? We address the questions in proposing conditions based on the study of functions variations. A particular focus is given to the relationship between strict sub-Gaussianity and symmetry of the distribution. In particular, we demonstrate that symmetry is neither sufficient nor necessary for strict sub-Gaussianity. In contrast, simple necessary conditions on the one hand, and simple sufficient conditions on the other hand, for strict sub-Gaussianity are provided. These results are illustrated via various applications to a number of bounded random variables, including Bernoulli, beta, binomial, Kumaraswamy, triangular, and uniform distributions.