TITLE: Distribution of $\max_{n \ge 0} S_n$, random walk
QUESTION [5 upvotes]: Say we have a random walk that is a nearest neighbor random walk on the integers where at each step the probability of moving one step to the right is $p$ and the probability of moving one step to the left is $q = 1-p$. Let $S_n$ be such a random walk started at $0$ for some $p \in (0, {1\over2})$. Let $M = \max_{n \ge 0} S_n$. What is the distribution of $M$?

REPLY [3 votes]: More generally, for any upward skip–free random walk on the integers (such that $P(X>1)=0$), the distribution of $M$ is geometric (when finite): $P(M=n)= (1-\theta) \cdot \theta^n$, where $\theta$ is the unique solution in $(0,1)$ of $1=E[\theta^{-X}]$.
The distribution of $M$ can also be calculated inductively for downward skip free random walk (such that $P(X<-1)=0$).
See e.g. Corollary 5.5 and Corollary 5.6 in Asmussen, S. (2003). Applied probability and queues, 2nd ed. New York: Springer, and http://people.bu.edu/pekoz/skipfree.pdf for other interesting information concerning skip free random walks.