Generalizations of Maximal Inequalities to Arbitrary Selection Rules

TL;DR

This paper extends classical maximal inequalities to arbitrary selection rules, providing tighter bounds and introducing new information-theoretic measures that could be useful beyond the scope of the inequalities.

Contribution

It generalizes maximal inequalities for arbitrary selection rules and introduces novel information-theoretic measures for tighter bounds.

Findings

Bounds are at least as tight as classical inequalities.

Bounds become significantly tighter when the selection index distribution is near deterministic.

New information-theoretic measures are introduced, potentially useful independently.

Abstract

We present a generalization of the maximal inequalities that upper bound the expectation of the maximum of $n$ jointly distributed random variables. We control the expectation of a randomly selected random variable from $n$ jointly distributed random variables, and present bounds that are at least as tight as the classical maximal inequalities, and much tighter when the distribution of selection index is near deterministic. A new family of information theoretic measures were introduced in the process, which may be of independent interest.