Probability inequalities and tail estimates for metric semigroups

TL;DR

This paper develops probability inequalities and tail estimates for sums of independent random variables in metric semigroups, extending classical results from Banach spaces to more general algebraic structures.

Contribution

It extends the Hoffmann-Jorgensen inequality to all metric semigroups, providing universal tail bounds and moment estimates in a broad mathematical framework.

Findings

Obtained tail estimates with universal constants.

Extended classical inequalities to metric semigroups.

Results apply to various algebraic structures including Lie groups.

Abstract

We study probability inequalities leading to tail estimates in a general semigroup $\mathscr{G}$ with a translation-invariant metric $d_{\mathscr{G}}$. (An important and central example of this in the functional analysis literature is that of $\mathscr{G}$ a Banach space.) Using our prior work [Ann. Prob. 2017] that extends the Hoffmann-Jorgensen inequality to all metric semigroups, we obtain tail estimates and approximate bounds for sums of independent semigroup-valued random variables, their moments, and decreasing rearrangements. In particular, we obtain the "correct" universal constants in several cases, extending results in the Banach space literature by Johnson-Schechtman-Zinn [Ann. Prob. 1985], Hitczenko [Ann. Prob. 1994], and Hitczenko and Montgomery-Smith [Ann. Prob. 2001]. Our results also hold more generally, in a very primitive mathematical framework required to state them: metric semigroups $\mathscr{G}$. This includes all compact, discrete, or (connected) abelian Lie groups.