The distribution of partial sums of random multiplicative functions with a large prime factor

Abstract: For $f$ a Steinhaus random multiplicative function, we prove convergence in distribution of the appropriately normalised partial sums [ \frac{{(\log \log x)}^{1/4}}{\sqrt{x}} \sum_{\substack{n \leq x \ P(n) > \sqrt{x}}} f(n), ] where $P(n)$ denotes the largest prime factor of $n$. We find that the limiting distribution is given by the square root of an integral with respect to a critical Gaussian multiplicative chaos measure multiplied by an independent standard complex normal random variable.