Probability inequalities for strongly left-invariant metric semigroups/monoids, including all lie groups

Recently, a general version of the Hoffmann-Jørgensen inequality was shown jointly with Rajaratnam [Ann. Probab. 2017], which (a) improved the result even for real-valued variables, but also (b) simultaneously unified and extended several versions in the Banach space literature, including that by Hitczenko–Montgomery-Smith [Ann. Probab. 2001], as well as special cases and variants of results by Johnson–Schechtman [Ann. Probab. 1989] and Klass–Nowicki [Ann. Probab. 2000], in addition to the original versions by Kahane and Hoffmann-Jørgensen. Moreover, our result with Rajaratnam was in a primitive framework: over all semigroups with a bi-invariant metric; this includes Banach spaces as well as compact and abelian Lie groups. In this note we show the result even more generally: over every semigroup \({\mathscr {G}}\) $\mathcal{G}$ with a strongly left- (or right-)invariant metric. We also prove some applications of this inequality over such \({\mathscr {G}}\) $\mathcal{G}$ , extending Banach space-valued versions by Hitczenko and Montgomery-Smith [Ann. Probab. 2001] and by Hoffmann-Jørgensen [Studia Math. 1974]. Furthermore, we show several other stochastic inequalities – by Ottaviani–Skorohod, Mogul’skii, and Lévy–Ottaviani – as well as Lévy’s equivalence, again over \({\mathscr {G}}\) $\mathcal{G}$ as above. This setting of generality for \({\mathscr {G}}\) $\mathcal{G}$ subsumes not only semigroups with bi-invariant metric (thus extending the previously shown results), but it also means that these results now hold over all Lie groups (equipped with a left-invariant Riemannian metric). We also explain why this primitive setting of strongly left/right-invariant metric semigroups \({\mathscr {G}}\) $\mathcal{G}$ is equivalent to that of left/right-invariant metric monoids \({\mathscr {G}}_\circ \) ${\mathcal{G}}_{\circ }$ : each such \({\mathscr {G}}\) $\mathcal{G}$ embeds in some \({\mathscr {G}}_\circ \) ${\mathcal{G}}_{\circ }$ .