Gaussian martingale inequality applies to random functions and maxima of empirical processes

TL;DR

This paper introduces a Bernstein-type Gaussian concentration inequality for martingales, improving existing bounds and applying to random functions, Erdős-Rényi graphs, and empirical process maxima.

Contribution

It presents a new Gaussian concentration inequality for martingales that enhances the Azuma-Hoeffding inequality and extends to various applications.

Findings

Improved Gaussian concentration bounds for martingales.

Applications to Erdős-Rényi random graphs.

Concentration results for maxima of empirical processes.

Abstract

We obtain a Bernstein type Gaussian concentration inequality for martingales. Our inequality improves the Azuma-Hoeffding inequality for moderate deviations $x$. Following the work of McDiarmid (1989), Talagrand (1996) and Boucheron, Lugosi and Massart (2000,2003), we show that our result can be applied to the concentration of random functions, Erd\"{o}s-R\'{e}nyi random graph, and maxima of empirical processes. Several interesting Gaussian concentration inequalities have been obtained.