A self-normalized law of the iterated logarithm for Rademacher random multiplicative functions

Abstract

Let $f$ be a Rademacher random multiplicative function supported on the squarefree integers, and let $M(x)=\sum_{n\le x}f(n)$. We prove a two-sided law of the iterated logarithm normalized by the full predictable quadratic variation \[ Q(x)=\sum_{p\le x}\bigg(\sum_{\substack{m\le x/p\\ P(m)<p}}f(m)\bigg)^2 \] of the martingale obtained by revealing the prime signs. Almost surely, \[ \limsup_{x\to\infty}\frac{M(x)}{\sqrt{2Q(x)\log\log x}}=1, \qquad \liminf_{x\to\infty}\frac{M(x)}{\sqrt{2Q(x)\log\log x}}=-1. \] The same equalities hold with $\log\log Q(x)$ in the denominator. The almost sure cluster set of the normalized sum is $[-1,1]$. This gives a self-normalized version of the questions of Wintner and Erdős on almost sure fluctuations. We also prove that $Q(x)/q(x)$ cannot converge in probability to $1$ for any positive deterministic function $q$. Moreover, \[ \limsup_{x\to\infty} \frac{|M(x)|}{\sqrt{x}(\log\log x)^{1/4}} \] is either an almost surely finite and positive random variable with nondegenerate distribution or infinite almost surely. We do not determine which alternative holds.