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 $\frac12$.
@article{10_37236_5330,
author = {Joseph Helfer and Daniel T. Wise},
title = {A note on maxima in random walks},
journal = {The electronic journal of combinatorics},
year = {2016},
volume = {23},
number = {1},
doi = {10.37236/5330},
zbl = {1331.60077},
url = {http://geodesic.mathdoc.fr/articles/10.37236/5330/}
}
TY - JOUR
AU - Joseph Helfer
AU - Daniel T. Wise
TI - A note on maxima in random walks
JO - The electronic journal of combinatorics
PY - 2016
VL - 23
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/5330/
DO - 10.37236/5330
ID - 10_37236_5330
ER -
%0 Journal Article
%A Joseph Helfer
%A Daniel T. Wise
%T A note on maxima in random walks
%J The electronic journal of combinatorics
%D 2016
%V 23
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/5330/
%R 10.37236/5330
%F 10_37236_5330
Joseph Helfer; Daniel T. Wise. A note on maxima in random walks. The electronic journal of combinatorics, Tome 23 (2016) no. 1. doi: 10.37236/5330
