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

TL;DR

This paper precisely determines the best constants in inequalities linking the sub-Gaussian norm and parameter for centered real-valued variables, with sharp bounds achieved by Gaussian and Rademacher distributions.

Contribution

It establishes the exact optimal constants in the inequalities relating sub-Gaussian norm and parameter, confirming their sharpness with specific distributions.

Findings

Bounds are .612 imes ext{sub-Gaussian norm} ext{ and } 0.832 imes ext{sub-Gaussian norm}

Bounds are sharp, attained by Gaussian and Rademacher distributions

Provides precise constants for sub-Gaussian inequalities

Abstract

We determine the optimal constants in the classical inequalities relating the sub-Gaussian norm \(\|X\|_{\psi_2}\) and the sub-Gaussian parameter \(\sigma_X\) for centered real-valued random variables. We show that \(\sqrt{3/8} \cdot \|X\|_{\psi_2} \le \sigma_X \le \sqrt{\log 2} \cdot \|X\|_{\psi_2}\), and that both bounds are sharp, attained by the standard Gaussian and Rademacher distributions, respectively.