On the limiting distribution of sums of random multiplicative functions

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.