Sharp Finite-Time Iterated-Logarithm Martingale Concentration

TL;DR

This paper develops sharp, finite-time concentration bounds for martingales that extend classical inequalities and establish a finite-time law of the iterated logarithm, providing insights into their relationship with the central limit theorem.

Contribution

It introduces optimal finite-time concentration bounds for martingales, extending Hoeffding and Bernstein inequalities, and proves a matching anti-concentration inequality.

Findings

Concentration bounds are optimal and uniform over finite times.

Established a finite-time version of the law of the iterated logarithm.

Connected the law of the iterated logarithm with the central limit theorem.

Abstract

We give concentration bounds for martingales that are uniform over finite times and extend classical Hoeffding and Bernstein inequalities. We also demonstrate our concentration bounds to be optimal with a matching anti-concentration inequality, proved using the same method. Together these constitute a finite-time version of the law of the iterated logarithm, and shed light on the relationship between it and the central limit theorem.