On the longest length of arithmetic progressions

Abstract

Suppose that ${\xi }_1^{(n)},{\xi }_2^{(n)},...,{\xi }_n^{(n)}$ are i.i.d with $P({\xi }_i^{(n)}=1)=p_n=1-P({\xi }_i^{(n)}=0)$. Let $U^{(n)}$ and $W^{(n)}$ be the longest length of arithmetic progressions and of arithmetic progressions mod $n$ relative to ${\xi }_1^{(n)},{\xi }_2^{(n)},...,{\xi }_n^{(n)}$ 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 ${\lim }_{n\to \infty }n{p}_n=\infty$. 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.

28 pages