A note on maxima in random walks

Joseph Helfer; Daniel T. Wise

doi:10.37236/5330

A note on maxima in random walks

Research output: Contribution to journal › Article › peer-review

Abstract

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.

Original language	English
Pages (from-to)	1-10
Number of pages	10
Journal	Electronic Journal of Combinatorics
Volume	23
Issue number	1
DOIs	https://doi.org/10.37236/5330
State	Published - 22 Jan 2016
Externally published	Yes

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
Geometry and Topology
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics
Applied Mathematics

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10.37236/5330

Cite this

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AB - 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.

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DO - 10.37236/5330

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A note on maxima in random walks

Abstract

All Science Journal Classification (ASJC) codes

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