  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 $\tfrac{1}{\sqrt{2}}$. 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
$\tfrac{1}{2}$.
