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 Khare and Rajaratnam (Ann Prob 45(6A):4101–4111, 2017) that extends the Hoffmann–Jørgensen 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 et al. (Ann Prob 13(1):234-253, 1985), Hitczenko (Ann Prob 22(1):453–468, 1994), and Hitczenko and Montgomery-Smith (Ann Prob 29(1):447-466, 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.