The peak sidelobe level of random binary sequences

TL;DR

This paper proves that the peak sidelobe level of a random binary sequence, normalized by actor, converges to actor , confirming a longstanding conjecture and settling a problem from the 1960s.

Contribution

It establishes the asymptotic behavior of the maximum autocorrelation of random binary sequences, confirming a conjecture and solving a problem from the 1960s.

Findings

$M(A_n)/\u221a{n\u03b4log n}$ converges in probability to actor

Expected value of $M(A_n)/actor$ tends to actor

Confirms a conjecture by Alon, Litsyn, and Shpunt

Abstract

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 $\sqrt{2}$.