Optimal sub-Gaussian variance proxy for truncated Gaussian and exponential random variables

Abstract

This paper establishes the optimal sub-Gaussian variance proxy for truncated Gaussian and truncated exponential random variables. The proofs are based initially on reducing each distribution to their standardized versions. Geometrically, for the normal distribution, our argument consists of fitting a parabola to another parabola-looking function, which emerges from its moment generating function. For the exponential case, we show that the optimal variance proxy is the unique solution to a pair of equations and then provide this solution explicitly. Moreover, we demonstrate that truncated Gaussian variables exhibit strict sub-Gaussian behavior if and only if they are symmetric, meaning their truncation is symmetric with respect to the mean. Conversely, truncated exponential variables are shown to never exhibit strict sub-Gaussianity.