An aperiodic binary sequence of length $\ell$ is a doubly infinite sequence $f=\ldots,f_{-1},f_0,f_1,\ldots$ 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 dot 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, skewness, and kurtosis of the distribution as a function of $\ell$. These revealed that for $\ell \geq 4$, the $p$th central moment of this distribution is strictly 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$\ell$的非周期二元序列是双无穷序列$f=\ldots,f_{-1},f_0,f_1,\ldots$，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需要那些不类似于自身平移的二元序列。$f$在平移量$s$处的自相关是$f$与$f$平移$s$位后序列的点积。$f$的劣化因子是归一化$f$至单位欧几里得范数后，所有非零平移自相关平方之和。因此，低劣化因子意味着低平移自相似性。我们赋予$2^\ell$个长度为$\ell$的二元序列均匀概率测度，并考虑其劣化因子的分布。早期工作利用组合技术给出了作为$\ell$函数的分布均值、方差、偏度和峰度的精确公式。这些结果揭示：对于$\ell \geq 4$，该分布的$p$阶中心矩对每个$p \geq 2$严格为正。本文证明每一$p$阶中心矩是$\ell$的拟多项式函数，其有理系数除以$\ell^{2p}$。同时证明，当$\ell$趋于无穷时，$p$阶标准化矩与标准正态分布的相应矩相同。