Probability inequalities and tail estimates for metric semigroups

Abstract

The goal of this work is to study probability inequalities leading to tail estimates in a general metric semigroup $\mathscr{G}$ with a translation-invariant metric $d_{\mathscr{G}}$. We begin by proving inequalities including those by Ottaviani-Skorohod, L\'evy, Mogul'skii, and Khinchin-Kahane in arbitrary semigroups $\mathscr{G}$. We then show a variant of Hoffmann-J{\o}rgensen's inequality, which unifies and significantly strengthens several versions in the Banach space literature, including those by Johnson and Schechtman [Ann. Prob. 17], Klass and Nowicki [Ann. Prob. 28], and Hitczenko and Montgomery-Smith [Ann. Prob. 29]. Moreover, our version of the inequality holds more generally, in the minimal mathematical framework of a metric semigroup $\mathscr{G}$. This inequality has important consequences (as in the Banach space literature) in obtaining 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, including in all normed linear spaces as well as in all compact, discrete, or abelian Lie groups.