A combinatorial approach to small ball inequalities for sums and differences

Understand

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 nonabelian 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.