On the longest length of arithmetic progressions

TL;DR

This paper investigates the asymptotic behavior of the longest arithmetic progressions and their modular counterparts in random binary sequences, providing distributional limits, error estimates, and almost sure convergence results.

Contribution

It derives the asymptotic distributions and almost sure limits of the longest arithmetic progressions in random sequences, including error bounds using Chen-Stein method.

Findings

Asymptotic distributions of $U^{(n)}$ and $W^{(n)}$ are established.

Error estimates are provided via Chen-Stein method.

Probability that $U^{(n)}$ and $W^{(n)}$ take small fixed values tends to 1 under certain conditions.

Abstract

Suppose that $\xi^{(n)}_1,\xi^{(n)}_2,...,\xi^{(n)}_n$ are i.i.d with $P(\xi^{(n)}_i=1)=p_n=1-P(\xi^{(n)}_i=0)$. Let $U^{(n)}$ and $W^{(n)}$ be the longest length of arithmetic progressions and of arithmetic progressions mod $n$ relative to $\xi^{(n)}_1,\xi^{(n)}_2,..., \xi^{(n)}_n$ respectively. Firstly, the asymptotic distributions of $U^{(n)}$ and $W^{(n)}$ are given. Simultaneously, the errors are estimated by using Chen-Stein method. Next, the almost surely limits are discussed when all $p_n$ are equal and when considered on a common probability space. Finally, we consider the case that $\lim_{n\to\infty}p_n=0$ and $\lim_{n\to\infty}{np_n}=\infty$. We prove that as $n$ tends to $\infty$, the probability that $U^{(n)}$ takes two numbers and $W^{(n)}$ takes three numbers tends to 1.