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 $\frac12$.
Highlights
A length walk is a sequence w : {1, . . . , } → {±1}
We give a combinatorial proof that a random walk attains a unique maximum with probability at least 1/2
We show that the probability that there is both a unique maximum and a unique minimum is asymptotically equal to
Summary
A length walk is a sequence w : {1, . . . , } → {±1}. The trajectory of w is the sequence w : {0, . . . , } → Z defined by w(j) =. We define max(w) = sup{w(j) : 0 j }. In [5], this is proven by a method that computes the probabilities of events in finite random walks by relating them to events in infinite random walks for which probabilities are more readily computed This is a general analytic method used to compute a large. 2n−r r−1 length 2n closed walks attaining their maximum exactly r times. 1 2 of attaining a unique maximum This conclusion does not assume that the walks are closed and allows an arbitrary distribution of step sizes.
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