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.

我们证明$k$- 区间上 Steinhaus 随机乘法函数的部分和的第一个正整数矩$（X,X+H]$匹配对应的高斯矩，只要$H\ll X∕{（\mathrm{日志}<!--FUNCTION\; APPLICATION-->X）}^{2k^2+2+哦（1）}$和$H$趋于无穷大$X$。我们证明，由随机乘法函数的实现所产生的典型乘法函数的适当归一化的部分和在短移动区间内具有高斯极限分布$（X,X+H]$和$H\ll X∕{（\mathrm{日志}<!--FUNCTION\; APPLICATION-->X）}^{瓦（X）}$趋于无穷大$X$， 在哪里$X$统一选自$\{1,2,\mathrm{\dots \dots }<!--FUNCTION\; APPLICATION-->,X\}$， 和$瓦（X）$趋于无穷大$X$随意地慢慢地。这使得哈珀最近的一个问题取得了一些初步进展。