On Orlicz spaces satisfying the Hoffmann-Jørgensen inequality

Abstract

Building on Talagrand's proof of the Hoffmann-Jørgensen inequality for $L_p$ spaces and its version for the exponential Orlicz spaces we provide a full characterization of Orlicz functions $\Psi$ for which an analogous inequality holds in the Orlicz space $L_{\Psi }(F)$, where $F$ is an arbitrary Banach space. As an application we present a characterization of Talagrand-type concentration inequality for suprema of empirical processes with envelope in $L_{\Psi }$ (equivalently for sums of independent $F$-valued random variables in $L_{\Psi }(F)$). This result generalizes in particular an inequality by the first-named author concerning exponentially integrable summands and a recent inequality due to Chamakh-Gobet-Liu on summands with $\beta$-heavy tails. Another corollary concerns concentration for convex functions of independent, unbounded random variables, generalizing recent results due to Klochkov-Zhivotovskiy and Sambale. We also obtain a corollary concerning boundedness in $L_{\Psi }(F)$ of partial sums of a series of independent random variables, generalizing the original result by Hoffmann-Jørgensen.