Annals of Probability

The Hoffmann–Jørgensen inequality in metric semigroups

Apoorva Khare and Bala Rajaratnam

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We prove a refinement of the inequality by Hoffmann–Jørgensen 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) 789–808], Klass and Nowicki [Ann. Probab. 28 (2000) 851–862], and Hitczenko and Montgomery-Smith [Ann. Probab. 29 (2001) 447–466]. Finally, we show that the Hoffmann–Jørgensen 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 $\mathscr{G}$. This includes normed linear spaces as well as all compact, discrete or (connected) abelian Lie groups.

Article information

Ann. Probab., Volume 45, Number 6A (2017), 4101-4111.

Received: May 2016
Revised: October 2016
First available in Project Euclid: 27 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization

Hoffmann–Jørgensen inequality metric semigroup


Khare, Apoorva; Rajaratnam, Bala. The Hoffmann–Jørgensen inequality in metric semigroups. Ann. Probab. 45 (2017), no. 6A, 4101--4111. doi:10.1214/16-AOP1160.

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