Distribution of random multiplicative functions in short intervals, with proper normalization

We determine the limiting distribution of partial sums of a Steinhaus random multiplicative function ${\sum }_{x\le n\le x+y}f(n)$ over short intervals $[x,x+y]$, where $y\to \mathrm{\infty }$ but $y=o(x)$. We show that with appropriate normalization, the limiting distribution is Gaussian for all such $y$. A key new feature of our result is that the normalization factor is different from the standard deviation $\sqrt{y}$ when $y$ is very close to $x$. In contrast, when $y\asymp x$ there is no normalization for which the limiting distribution is a non-degenerate Gaussian.