A Note on Maxima in Random Walks

Abstract

We give a combinatorial proof that a random walk attains a unique maximum with probability at least $1/2$. For closed random walks with uniform step size, we recover Dwass's count of the number of length $\ell$ walks attaining the maximum exactly $k$ times. We also show that the probability that there is both a unique maximum and a unique minimum is asymptotically equal to $\frac14$ and that the probability that a Dyck word has a unique minimum is asymptotically $\frac12$.