This paper establishes the optimal sub-Gaussian variance proxy for truncated Gaussian and truncated exponential random variables. The proofs rely on first characterizing the optimal variance proxy as the unique solution to a set of two equations and then observing that for these two truncated distributions, one may find explicit solutions to this set of equations. Moreover, we establish the conditions under which the optimal variance proxy coincides with the variance, thereby characterizing the strict sub-Gaussianity of the truncated random variables. Specifically, 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-Gaussian properties. These findings contribute to the understanding of these prevalent probability distributions in statistics and machine learning, providing a valuable foundation for improved and optimal modeling and decision-making processes.

翻译：本文确立了截断高斯和截断指数型随机变量的最优次高斯方差代理。证明过程首先将最优方差代理刻画为一组两个方程的唯一解，随后发现对于这两种截断分布，该方程组存在显式解。此外，本文确定了最优方差代理与方差相等的条件，进而刻画了截断随机变量的严格次高斯性。具体而言，我们证明截断高斯变量在且仅当其关于均值对称截断时才表现出严格次高斯行为；相反，截断指数型变量被证明永远不满足严格次高斯性质。这些发现有助于理解统计学与机器学习中这些常用概率分布，为改进和优化建模与决策过程提供了重要基础。