Concentration inequalities for randomly permuted sums

TL;DR

This paper introduces a new proof technique for Bernstein-type concentration inequalities for permuted sums, enhancing understanding of permutation test properties and their Gaussian behavior.

Contribution

It provides a novel proof based on Talagrand's inequalities, improving bounds and understanding of concentration for permuted sums.

Findings

New proof of Bernstein-type inequality using Talagrand's inequalities

Reveals Gaussian behavior of permuted sums under classical conditions

Application to permutation tests of independence and error rate analysis

Abstract

Initially motivated by the study of the non-asymptotic properties of non-parametric tests based on permutation methods, concentration inequalities for uniformly permuted sums have been largely studied in the literature. Recently, Delyon et al. proved a new Bernstein-type concentration inequality based on martingale theory. This work presents a new proof of this inequality based on the fundamental inequalities for random permutations of Talagrand. The idea is to first obtain a rough inequality for the square root of the permuted sum, and then, iterate the previous analysis and plug this first inequality to obtain a general concentration of permuted sums around their median. Then, concentration inequalities around the mean are deduced. This method allows us to obtain the Bernstein-type inequality up to constants, and, in particular, to recovers the Gaussian behavior of such permuted sums under classical conditions encountered in the literature. Then, an application to the study of the second kind error rate of permutation tests of independence is presented.