A Systematic Martingale Construction with Applications to Permutation Inequalities

TL;DR

This paper introduces a systematic method for constructing martingales from sampling processes and applies it to derive new and existing permutation inequalities, enhancing understanding of sequence rearrangements.

Contribution

It presents a novel systematic martingale construction method and applies it to develop new permutation inequalities and variations of classical results.

Findings

New martingale constructions from sampling without replacement

Development of maximal inequalities for permuted sequences

Extensions and variations of classical permutation inequalities

Abstract

We illustrate a process that constructs martingales from raw material that arises naturally from the theory of sampling without replacement.The usefulness of the new martingales is illustrated by the development of maximal inequalities for permuted sequences of real numbers. Some of these inequalities are new and some are variations of classical inequalities like those introduced by A. Garsia in the study of rearrangement of orthogonal series.