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A Combinatorial Approach to Small Ball Inequalities for Sums and Differences

Part of: Probability theory on algebraic and topological structures Combinatorial probability Extremal combinatorics

Published online by Cambridge University Press:  06 November 2018

JIANGE LI  and
MOKSHAY MADIMAN
JIANGE LI
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA (e-mail: lijiange@udel.edu, madiman@udel.edu)
MOKSHAY MADIMAN
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA (e-mail: lijiange@udel.edu, madiman@udel.edu)
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Abstract

Small ball inequalities have been extensively studied in the setting of Gaussian processes and associated Banach or Hilbert spaces. In this paper, we focus on studying small ball probabilities for sums or differences of independent, identically distributed random elements taking values in very general sets. Depending on the setting – abelian or non-abelian groups, or vector spaces, or Banach spaces – we provide a collection of inequalities relating different small ball probabilities that are sharp in many cases of interest. We prove these distribution-free probabilistic inequalities by showing that underlying them are inequalities of extremal combinatorial nature, related among other things to classical packing problems such as the kissing number problem. Applications are given to moment inequalities.

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 05D05: Extremal set theory 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Issue 1 , January 2019 , pp. 100 - 129
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

†

This work was supported in part by the US National Science Foundation through grants CCF-1346564 and DMS-1409504 (CAREER)

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