On the longest length of arithmetic progressions

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.