A note on maxima in random walks

We give a combinatorial proof that a random walk attains a unique maximum with probability at least ½. For closed random walks with uniform step size, we recover Dwass’s count of the number of length ℓ 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 ¼ and that the probability that a Dyck word has a unique minimum is asymptotically ½.