Counting sign changes of partial sums of random multiplicative functions

ABSTRACT 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 construct a family of increasing intervals $[y_k, X_k]$ that contain at least 1 sign change of $M_f(u)$ eventually, almost surely. As a consequence, we show that for any constant c > 2, $$ V_f(x) = \Omega ((\log \log \log x)^{1/c} ) $$ almost surely.