Open Access
November 2017 The Hoffmann–Jørgensen inequality in metric semigroups
Apoorva Khare, Bala Rajaratnam
Ann. Probab. 45(6A): 4101-4111 (November 2017). DOI: 10.1214/16-AOP1160


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.


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Apoorva Khare. Bala Rajaratnam. "The Hoffmann–Jørgensen inequality in metric semigroups." Ann. Probab. 45 (6A) 4101 - 4111, November 2017.


Received: 1 May 2016; Revised: 1 October 2016; Published: November 2017
First available in Project Euclid: 27 November 2017

zbMATH: 06838116
MathSciNet: MR3729624
Digital Object Identifier: 10.1214/16-AOP1160

Primary: 60E15
Secondary: 60B15

Keywords: Hoffmann–Jørgensen inequality , metric semigroup

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.45 • No. 6A • November 2017
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