scholarly journals A Gaussian upper bound for martingale small-ball probabilities

2016 ◽  
Vol 44 (6) ◽  
pp. 4184-4197
Author(s):  
James R. Lee ◽  
Yuval Peres ◽  
Charles K. Smart
Keyword(s):  
Upper Bound ◽  
Small Ball ◽  
Small Ball Probabilities

scholarly journals A Combinatorial Approach to Small Ball Inequalities for Sums and Differences

2018 ◽  
Vol 28 (1) ◽  
pp. 100-129 ◽  
Author(s):  
JIANGE LI ◽  
MOKSHAY MADIMAN
Keyword(s):  
Hilbert Spaces ◽  
Vector Spaces ◽  
Combinatorial Approach ◽  
Kissing Number ◽  
Small Ball ◽  
Small Ball Probabilities ◽  
Packing Problems ◽  
Number Problem ◽  
Probabilistic Inequalities ◽  
Independent Identically Distributed

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.

scholarly journals Small-Ball Probabilities for the Volume of Random Convex Sets

2013 ◽  
Vol 49 (3) ◽  
pp. 601-646 ◽  
Author(s):  
Grigoris Paouris ◽  
Peter Pivovarov
Keyword(s):  
Convex Sets ◽  
Small Ball ◽  
Small Ball Probabilities ◽  
Random Convex
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