  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.
