Probability inequalities and tail estimates for metric semigroups

Abstract

We study probability inequalities leading to tail estimates in a general semigroup $\mathcal{G}$ with a translation-invariant metric $d_{\mathcal{G}}$. (An important and central example of this in the functional analysis literature is that of $\mathcal{G}$ 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 $\mathcal{G}$. This includes all compact, discrete, or (connected) abelian Lie groups.

13 pages, final version, published in Advances in Operator Theory

References in corpus (1)

• The Hoffmann-Jorgensen inequality in metric semigroups

Cited by in corpus (3)

• Differential calculus on the space of countable labelled graphs
• The Khinchin-Kahane and Levy inequalities for abelian metric groups, and transfer from normed (abelian semi)groups to Banach spaces
• Probability inequalities for strongly left-invariant metric semigroups/monoids, including all Lie groups