Random multiplicative functions in short intervals

TL;DR

This paper proves a central limit theorem for sums of random multiplicative functions in short intervals using Stein's method, advancing understanding of their probabilistic behavior.

Contribution

It introduces a novel application of Stein's method to establish normal approximation for sums of random multiplicative functions in short intervals.

Findings

Established a CLT for random multiplicative functions in short intervals

Demonstrated the effectiveness of Stein's method in this context

Provided quantitative bounds on the approximation

Abstract

We consider random multiplicative functions taking the values $\pm 1$. Using Stein's method for normal approximation, we prove a central limit theorem for the sum of such multiplicative functions in appropriate short intervals.