  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 $H\ll x/(\log x)^{2k^2+2+o(1)}$ 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 $H\ll X/(\log X)^{W(X)}$ tending to infinity with
$X$, where $x$ is uniformly chosen from $\{1,2,\dots, X\}$, and $W(X)$ tends to
infinity with $X$ arbitrarily slowly. This makes some initial progress on a
recent question of Harper.
