On the Maximum of a Simple Random Walk

Abstract

Let $S_0 = 0$, $S_n = \xi _1 + \xi _2 + \cdots + \xi _n $, $n \geq 1$, be the simple random walk generated by a sequence of independent random variables $\xi _i $, $i = 1,2, \ldots $, such that ${\bf P}\{ {\xi _i = 1} \} = 1 - {\bf P}\{ {\xi _i = - 1} \} = \tfrac{1}{2}$, and let T be the moment of the first return of $S_n $ to the state 0. We find an asymptotic representation for the probability ${\bf P}\{ {\max _{0 < k < T} |S_k | > n|T = 2N} \}$ which is exact (in order), assuming that $n^2 N^{ - 1} \to \infty $, and $nN^{ - 1} \leq a < 1$. The results obtained are used to study the asymptotics of moderate and large deviations of the height of a planted plane tree with N vertices.