Probability inequalities for strongly left-invariant metric semigroups/monoids, including all lie groups

Khare, A (2024) Probability inequalities for strongly left-invariant metric semigroups/monoids, including all lie groups. In: Indian Journal of Pure and Applied Mathematics .

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Abstract

Recently, a general version of the Hoffmann-Jørgensen inequality was shown jointly with Rajaratnam Ann. Probab. 2017, which (a) improved the result even for real-valued variables, but also (b) simultaneously unified and extended several versions in the Banach space literature, including that by Hitczenko�Montgomery-Smith Ann. Probab. 2001, as well as special cases and variants of results by Johnson�Schechtman Ann. Probab. 1989 and Klass�Nowicki Ann. Probab. 2000, in addition to the original versions by Kahane and Hoffmann-Jørgensen. Moreover, our result with Rajaratnam was in a primitive framework: over all semigroups with a bi-invariant metric; this includes Banach spaces as well as compact and abelian Lie groups. In this note we show the result even more generally: over every semigroup G with a strongly left- (or right-)invariant metric. We also prove some applications of this inequality over such G, extending Banach space-valued versions by Hitczenko and Montgomery-Smith Ann. Probab. 2001 and by Hoffmann-Jørgensen Studia Math. 1974. Furthermore, we show several other stochastic inequalities � by Ottaviani�Skorohod, Mogul�skii, and Lévy�Ottaviani � as well as Lévy�s equivalence, again over G as above. This setting of generality for G subsumes not only semigroups with bi-invariant metric (thus extending the previously shown results), but it also means that these results now hold over all Lie groups (equipped with a left-invariant Riemannian metric). We also explain why this primitive setting of strongly left/right-invariant metric semigroups G is equivalent to that of left/right-invariant metric monoids G�: each such G embeds in some G�. © The Indian National Science Academy 2024.

Item Type:	Journal Article
Publication:	Indian Journal of Pure and Applied Mathematics
Publisher:	Indian National Science Academy
Additional Information:	The copyright for this article belongs to Indian National Science Academy.
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	18 Dec 2024 12:25
Last Modified:	18 Dec 2024 12:25
URI:	http://eprints.iisc.ac.in/id/eprint/85893

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