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

2025/07/08 by Lasse Leskelä, Leskelä, Lasse, Matvei Zhukov +1 · 1 citation
Decision Sciences · Mathematics · #60E05 #60E15 #FOS: Mathematics #Geometry and complex manifolds #Probability (math.PR) #Probability and Risk Models #Random Matrices and Applications #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2507.05928

openalex publication_date 2025/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We determine the optimal constants in the classical inequalities relating the sub-Gaussian norm \(‖X‖_ψ2\) and the sub-Gaussian parameter \(σ_X\) for centered real-valued random variables. We show that \(√(3/8) ⋅ ‖X‖_ψ2 ≤ σ_X ≤ √(log 2) ⋅ ‖X‖_ψ2\), and that both bounds are sharp, attained by the standard Gaussian and Rademacher distributions, respectively.

Citations

• A Simple Note on the Basic Properties of Subgaussian Random Variables
• Stochastic order characterization of uniform integrability and tightness
• Extreme Points of Moment Sets

Cited by

• Sub-Poisson distributions: Concentration inequalities, optimal variance proxies, and closure properties

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