Martingale central limit theorem for random multiplicative functions

Let α be a Steinhaus or a Rademacher random multiplicative function. For a wide class of multiplicative functions f we prove that the sum ∑n ≤ xα(n) f(n), normalised to have mean square $1$, has a limiting distribution. The limiting distribution we find is a Gaussian times the square-root of the total mass of a random measure associated with f. Our result applies to dz, the z-th divisor function, as long as z is strictly between $0$ and 1√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 1/2.