An aperiodic binary sequence of length $\ell$ is written as $f=\ldots,f_{-1},f_0,f_1,\ldots$ with $f_j \in \{-1,1\}$ when $0 \leq j < \ell$ and and $f_j=0$ otherwise. Various problems in engineering and natural science demand binary sequences that do not resemble translates of themselves. The autocorrelation of $f$ at shift $s$ is the inner product of $f$ with the sequence obtained by translating $f$ by $s$ places. The demerit factor of $f$ is the sum of the squares of the autocorrelations at all nonzero shifts for the sequence obtained by normalizing $f$ to unit Euclidean norm. Low demerit factor therefore indicates low self-similarity under translation. We endow the $2^\ell$ binary sequences of length $\ell$ with uniform probability measure and consider the distribution of their demerit factors. Earlier works used combinatorial techniques to find exact formulas for the mean, variance, and skewness of the distribution as a function of $\ell$. These revealed that for $\ell \geq 4$, the $p$th central moment of this distribution is positive for every $p \geq 2$. This article shows that every $p$th central moment is a quasi-polynomial function of $\ell$ with rational coefficients divided by $\ell^{2 p}$. It also shows that, in the limit as $\ell$ tends to infinity, the $p$th standardized moment is the same as that of the standard normal distribution.

翻译：长度为$\ell$的非周期二进制序列记为$f=\ldots,f_{-1},f_0,f_1,\ldots$，其中当$0 \leq j < \ell$时$f_j \in \{-1,1\}$，否则$f_j=0$。工程与自然科学中的诸多问题要求二进制序列不近似于自身的平移。$f$在位移$s$处的自相关定义为$f$与将$f$平移$s$位所得序列的内积。$f$的劣化因子是将$f$归一化至单位欧几里得范数后，所有非零位移处自相关平方之和。因此，低劣化因子表明序列在平移下具有低自相似性。我们赋予$2^\ell$个长度为$\ell$的二进制序列均匀概率测度，并考虑其劣化因子的分布。早期工作利用组合方法得到了该分布的均值、方差和偏度作为$\ell$函数的精确公式。这些结果表明，当$\ell \geq 4$时，该分布的$p$阶中心矩对于所有$p \geq 2$均为正。本文证明，每个$p$阶中心矩均为$\ell$的拟多项式函数，其有理系数除以$\ell^{2p}$。此外，在$\ell$趋于无穷的极限下，$p$阶标准化矩与标准正态分布的标准化矩相同。