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.
Categories: math.PR