The Hoffmann-Jorgensen inequality in metric semigroups

Abstract

We prove a refinement of the inequality by Hoffmann-Jorgensen that is significant for three reasons. First, our result improves on the state-of-the-art even for real-valued random variables. Second, the result unifies several versions in the Banach space literature, including those by Johnson and Schechtman [Ann. Probab. 17 (1989)], Klass and Nowicki [Ann. Probab. 28 (2000)], and Hitczenko and Montgomery-Smith [Ann. Probab. 29 (2001)]. Finally, we show that the Hoffmann-Jorgensen inequality (including our generalized version) holds not only in Banach spaces but more generally, in a very primitive mathematical framework required to state the inequality: a metric semigroup $\mathcal{G}$. This includes normed linear spaces as well as all compact, discrete, or (connected) abelian Lie groups.

11 pages, published in the Annals of Probability. The Introduction section shares motivating examples with arXiv:1506.02605

References in corpus (2)

• Integration and measures on the space of countable labelled graphs
• Differential calculus on the space of countable labelled graphs

Cited by in corpus (5)

• Homogeneous length functions on groups
• Probability inequalities and tail estimates for metric semigroups
• Differential calculus on the space of countable labelled graphs
• The Khinchin-Kahane and Levy inequalities for abelian metric groups, and transfer from normed (abelian semi)groups to Banach spaces
• Probability inequalities for strongly left-invariant metric semigroups/monoids, including all Lie groups