On the longest length of arithmetic progressions

MinZhi Zhao, Huizeng Zhang

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…