Martingale central limit theorem for random multiplicative functions

Abstract

Let $\alpha$ be a Steinhaus or a Rademacher random multiplicative function. For a wide class of multiplicative functions $f$ we show that the sum ${\sum }_{n\le x}\alpha (n)f(n)$, normalised to have mean square $1$, has a non-Gaussian limiting distribution. More precisely, we establish a generalised central limit theorem with random variance determined by the total mass of a random measure associated with $\alpha f$. Our result applies to $d_z$, the $z$-th divisor function, as long as $z$ is strictly between $0$ and $\frac{1}{\sqrt{2}}$. Other examples of admissible $f$-s include any multiplicative indicator function with the property that $f(p)=1$ holds for a set of primes of density strictly between $0$ and $\frac{1}{2}$.

42 pages, 2 figures. Typos fixed, abstract and discussion of previous works updated. Codes for simulation experiment available on authors' personal page; comments are still welcome