随机二元序列的峰值旁瓣电平
The peak sidelobe level of random binary sequences
摘要：
设$A_n=(a_0,a_1,\dots,a_{n-1})$是从$\{-1,+1\}^n$均匀随机抽取的，定义了\[ M(A_n)=\max_{0<u<n}\,\Bigg|\sum_{j=0}^{n-u-1}a_ja_{j+u}\Bigg|\quad\text{for $n>1$}. \]，证明了$M(A_n)/\sqrt{n\log n}$依概率收敛于$\sqrt{2}$ .这解决了Moon和Moser在20世纪60年代首次研究的一个问题，并肯定地证明了Alon，Litsyn和Shpunt最近提出的一个猜想。研究还表明，$M(A_n)/\sqrt{n\log n}$的期望值趋于$\sqrt{2}$ .
英文摘要：
Let $A_n=(a_0,a_1,\dots,a_{n-1})$ be drawn uniformly at random from $\{-1,+1\}^n$ and define \[ M(A_n)=\max_{0<u<n}\,\Bigg|\sum_{j=0}^{n-u-1}a_ja_{j+u}\Bigg|\quad\text{for $n>1$}. \] It is proved that $M(A_n)/\sqrt{n\log n}$ converges in probability to $\sqrt{2}$. This settles a problem first studied by Moon and Moser in the 1960s and proves in the affirmative a recent conjecture due to Alon, Litsyn, and Shpunt. It is also shown that the expectation of $M(A_n)/\sqrt{n\log n}$ tends to $\sqrt{2}$.
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