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

Mathias Barreto (1) , Marchal Olivier (2, 3, 4, 5) , Julyan Arbel (6)

1 HSE - Vysšaja škola èkonomiki = National Research University Higher School of Economics [Moscow]
2 ICJ - Institut Camille Jordan
3 PSPM - Probabilités, statistique, physique mathématique
4 IUF - Institut universitaire de France
5 UJM FST - Université Jean Monnet - Faculté des Sciences et Techniques
6 STATIFY - Modèles statistiques bayésiens et des valeurs extrêmes pour données structurées et de grande dimension

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.

Mots clés: Sub-Gaussian random variables; Optimal variance proxy; Truncated distributions

Domaines: Probabilités [math.PR]

Mathias Barreto, Marchal Olivier, Julyan Arbel. Optimal sub-Gaussian variance proxy for truncated Gaussian and exponential random variables. 2024. ⟨hal-04503494⟩