Concentration inequalities for strong laws and laws of the iterated logarithm

TL;DR

This paper develops concentration inequalities that provide non-asymptotic bounds for sums of i.i.d. random variables, extending classical strong laws and laws of the iterated logarithm with new inequalities and pathwise analogues.

Contribution

It introduces novel concentration inequalities that generalize classical results on strong laws and laws of the iterated logarithm in a non-asymptotic framework.

Findings

Derived non-asymptotic inequalities for strong laws of large numbers.

Established pathwise analogues of laws of the iterated logarithm.

Extended classical results with new concentration bounds.

Abstract

We derive concentration inequalities for sums of independent and identically distributed random variables that yield non-asymptotic generalizations of several strong laws of large numbers including some of those due to Kolmogorov [1930], Marcinkiewicz and Zygmund [1937], Chung [1951], Baum and Katz [1965], Ruf, Larsson, Koolen, and Ramdas [2023], and Waudby-Smith, Larsson, and Ramdas [2024]. As applications, we derive non-asymptotic iterated logarithm inequalities in the spirit of Darling and Robbins [1967], as well as pathwise (sometimes described as "game-theoretic") analogues of strong laws and laws of the iterated logarithm.