  Let $f$ be a Rademacher random multiplicative function. Let $$M_f(u):=\sum_{n
\leq u} f(n)$$ be the partial sum of $f$. Let $V_f(x)$ denote the number of
sign changes of $M_f(u)$ up to $x$. We show that for any constant $c > 2$,
$$V_f(x) = \Omega ((\log \log \log x)^{1/c} )$$ almost surely.
