$\Phi$-moment inequalities for independent and freely independent random variables

TL;DR

This paper establishes new $ ext{ extbackslash Phi}$-moment inequalities for sums of independent and freely independent random variables, providing equivalent expressions and characterizations in classical and free probability settings.

Contribution

It introduces novel $ ext{ extbackslash Phi}$-moment inequalities and equivalent characterizations for sums and maxima of independent and freely independent variables, extending classical results to free probability.

Findings

Equivalent expression for $ ext{ extbackslash E}( ext{ extbackslash Phi}( ext{ extbackslash sum} f_k))$ in terms of disjoint copies.

Characterization of $ au( ext{ extbackslash Phi}( ext{ extbackslash sup}^+ x_k))$ for freely independent variables.

New results on free Johnson-Schechtman inequalities in quasi-Banach symmetric operator spaces.

Abstract

This paper is devoted to the study of $\Phi$-moments of sums of independent/freely independent random variables. More precisely, let $(f_k)_{k=1}^n$ be a sequence of positive (symmetrically distributed) independent random variables and let $\Phi$ be an Orlicz function with $\Delta_2$-condition. We provide an equivalent expression for the quantity $\mathbb{E}(\Phi(\sum_{k=1}^n f_k))$ in term of the sum of disjoint copies of the sequence $(f_k)_{k=1}^n.$ We also prove an analogous result in the setting of free probability. Furthermore, we provide an equivalent characterization of $\tau(\Phi(\sup^+_{1\leq k\leq n}x_k))$ for positive freely independent random variables and also present some new results on free Johnson-Schechtman inequalities in the quasi-Banach symmetric operator space.