  We study two models of random multiplicative functions: Rademacher random
multiplicative functions supported on the squarefree integers $f$, and
Rademacher random completely multiplicative functions $f^*$. We prove that the
partial sums $\sum_{n\leq x}f^*(n)$ and $\sum_{n\leq x}\frac{f(n)}{\sqrt{n}}$
change sign infinitely often as $x\to\infty$, almost surely. The case
$\sum_{n\leq x}\frac{f^*(n)}{\sqrt{n}}$ is left as an open question and we
stress the possibility of only a finite number of sign changes, with positive
probability.
