The Hoffmann-Jorgensen inequality in metric semigroups

TL;DR

This paper presents a refined Hoffmann-Jorgensen inequality applicable in metric semigroups, improving existing bounds for real-valued variables and unifying various Banach space results within a broader mathematical framework.

Contribution

It introduces a generalized inequality valid in metric semigroups, extending previous results from Banach spaces and real-valued variables, and unifies multiple versions in the literature.

Findings

Improved bounds for real-valued random variables.

Unified framework encompassing several Banach space inequalities.

Validity of the inequality in general metric semigroups, including Lie groups.

Abstract

We prove a refinement of the inequality by Hoffmann-Jorgensen that is significant for three reasons. First, our result improves on the state-of-the-art even for real-valued random variables. Second, the result unifies several versions in the Banach space literature, including those by Johnson and Schechtman [Ann. Probab. 17 (1989)], Klass and Nowicki [Ann. Probab. 28 (2000)], and Hitczenko and Montgomery-Smith [Ann. Probab. 29 (2001)]. Finally, we show that the Hoffmann-Jorgensen inequality (including our generalized version) holds not only in Banach spaces but more generally, in a very primitive mathematical framework required to state the inequality: a metric semigroup $\mathscr{G}$. This includes normed linear spaces as well as all compact, discrete, or (connected) abelian Lie groups.