Strong Approximations in the Almost Sure Central Limit Theorem and Limit Behavior of the Center of Mass

TL;DR

This paper proves almost sure central limit theorems and laws of the iterated logarithm for random sequences and their centers of mass, with applications to reinforced random walks.

Contribution

It introduces new strong approximation conditions enabling almost sure convergence results and extends classical limit theorems to the center of mass of random sequences.

Findings

Established almost sure CLT under strong approximation conditions

Derived law of the iterated logarithm for the center of mass

Applied results to step-reinforced random walks

Abstract

In this paper, we establish an almost sure central limit theorem for a general random sequence under a strong approximation condition. Additionally, we derive the law of the iterated logarithm for the center of mass corresponding to a random sequence under a different strong approximation condition. Applications to step-reinforced random walks are also discussed.