On the limiting distribution of sums of random multiplicative functions

TL;DR

This paper determines the limiting distribution of sums of Steinhaus random multiplicative functions, using advanced probabilistic techniques and novel conditioning methods to handle criticality and convergence issues.

Contribution

It introduces a new approach to analyze the distribution of multiplicative functions at criticality, extending previous work with innovative conditioning and coupling techniques.

Findings

Established the limiting distribution of the scaled sum of Steinhaus functions.

Proved convergence of associated Euler products to a critical multiplicative chaos measure.

Developed a novel coupling and homogenisation method for universality of non-Gaussian chaos.

Abstract

We establish the limiting distribution of $\frac{{(\log \log x)}^{1/4}}{\sqrt{x}} \sum_{n\le x}\alpha(n)$ where $\alpha$ is a Steinhaus random multiplicative function, answering a question of Harper. The distributional convergence is proved by applying the martingale central limit theorem to a suitably truncated sum. This truncation is inspired by work of Najnudel, Paquette, Simm and Vu on subcritical holomorphic multiplicative chaos setting, but analysed with a different conditioning argument generalised from Harper's work on fractional moments to circumvent integrability issues at criticality. A significant part of the proof is devoted to the convergence in probability of the associated partial Euler product to a critical multiplicative chaos measure, independent of the mild shift away from the critical line. Our approach to the universality of critical non-Gaussian multiplicative chaos bypasses the barrier analysis with the help of a modified second moment method, and employs a novel argument based on coupling and homogenisation by change of measure, which could be of independent interest.