A Note on Maxima in Random Walks

  • Joseph Helfer
  • Daniel T. Wise
Keywords: Dyck words, Dyck paths, Random walks, Catalan numbers


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 $\frac12$.

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