Partial sums of typical multiplicative functions over short moving intervals

We prove that the $k$-th positive integer moment of partial sums of Steinhaus random multiplicative functions over the interval $(x,x+H]$ matches the corresponding Gaussian moment, as long as $H\ll x∕{(\log <!--FUNCTION\; APPLICATION-->x)}^{2k^2+2+o(1)}$ and $H$ tends to infinity with $x$. We show that properly normalized partial sums of typical multiplicative functions arising from realizations of random multiplicative functions have Gaussian limiting distribution in short moving intervals $(x,x+H]$ with $H\ll X∕{(\log <!--FUNCTION\; APPLICATION-->X)}^{W(X)}$ tending to infinity with $X$, where $x$ is uniformly chosen from $\{1,2,\dots <!--FUNCTION\; APPLICATION-->,X\}$, and $W(X)$ tends to infinity with $X$ arbitrarily slowly. This makes some initial progress on a recent question of Harper.