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arXiv · 1610.02324

The Hoffmann-Jorgensen inequality in metric semigroups

Apoorva Khare·Bala Rajaratnam
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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 $\mathscr{G}$. This includes normed linear spaces as well as all compact, discrete, or (connected) abelian Lie groups.

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Apoorva Khare, Bala Rajaratnam. 2016-10-07. The Hoffmann-Jorgensen inequality in metric semigroups. https://doi.org/10.1214/16-aop1160

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