Sharp constants relating the sub-Gaussian norm and the sub-Gaussian parameter

Abstract

We determine the optimal constants in the classical inequalities relating the sub-Gaussian norm ‖X‖ψ₂ and the sub-Gaussian parameter σX for centered real-valued random variables. We show that (3/8)1/2 · ‖X‖ψ₂ ≤ σX ≤ (log 2)1/2 · ‖X‖ψ₂, and that both bounds are sharp, attained by the standard Gaussian and Rademacher distributions, respectively.