Probability inequalities over left-invariant metric semigroups

Monday, October 28, 2024, 10:00 -- 10:30

We explore how a classical stochastic inequality named after Hoffmann-Jorgensen extends beyond the Banach space setting. This inequality can be stated using only a binary associative operation and a distance function. Thus, we present a generalization that (i) holds in a very primitive mathematical setting: left-invariant metric monoids. In particular, our version holds over every Lie group, in addition to all Banach spaces. The result moreover (ii) strictly strengthens prior versions (even for real variables), and (iii) unifies previously disparate Banach space variants in the literature. We will also see some consequences.