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arXiv · 2507.05928

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

Lasse Leskelä·Matvei Zhukov
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Abstract

We determine the optimal constants in the classical inequalities relating the sub-Gaussian norm \(\|X\|_{ψ_2}\) and the sub-Gaussian parameter \(σ_X\) for centered real-valued random variables. We show that \(\sqrt{3/8} \cdot \|X\|_{ψ_2} \le σ_X \le \sqrt{\log 2} \cdot \|X\|_{ψ_2}\), and that both bounds are sharp, attained by the standard Gaussian and Rademacher distributions, respectively.

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Lasse Leskelä, Matvei Zhukov. 2025-07-08. Sharp constants relating the sub-Gaussian norm and the sub-Gaussian parameter. https://arxiv.org/abs/2507.05928

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