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 $\ell$ 的非周期二进制序列是一个双无限序列 $f=\ldots,f_{-1},f_0,f_1,\ldots$，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各类问题需要那些与自身平移副本不相似的二进制序列。序列 $f$ 在平移量 $s$ 处的自相关是 $f$ 与将 $f$ 平移 $s$ 位后所得序列的点积。$f$ 的减损因子是将 $f$ 归一化为单位欧几里得范数后所得序列在所有非零平移量处自相关平方之和。因此，较低的减损因子表明序列在平移下具有较低的自相似性。我们对所有 $2^\ell$ 个长度为 $\ell$ 的二进制序列赋予均匀概率测度，并研究其减损因子的分布。先前的研究使用组合技术得到了该分布的均值、方差、偏度和峰度关于 $\ell$ 的精确公式。这些结果表明，对于 $\ell \geq 4$，该分布的 $p$ 阶中心矩对每个 $p \geq 2$ 均为严格正值。本文证明了每一 $p$ 阶中心矩均为 $\ell$ 的拟多项式函数，其有理系数除以 $\ell^{2 p}$。同时证明了当 $\ell$ 趋于无穷大时，$p$ 阶标准化矩与标准正态分布的相应矩相同。