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The Maximum of a Symmetric Next Neighbor Walk on the Nonnegative Integers

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Ora E. Percus  and
Jerome K. Percus
Ora E. Percus*
Affiliation:
New York University
Jerome K. Percus*
Affiliation:
New York University
*
∗ Postal address: Courant Institute, New York University, 251 Mercer Street, New York, NY 10012, USA.
∗ Postal address: Courant Institute, New York University, 251 Mercer Street, New York, NY 10012, USA.
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Abstract

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We consider a one-dimensional discrete symmetric random walk with a reflecting boundary at the origin. Generating functions are found for the two-dimensional probability distribution P{Sn = x, max1≤j≤nSn = a} of being at position x after n steps, while the maximal location that the walker has achieved during these n steps is a. We also obtain the familiar (marginal) one-dimensional distribution for Sn = x, but more importantly that for max1≤j≤nSj = a asymptotically at fixed a2 / n. We are able to compute and compare the expectations and variances of the two one-dimensional distributions, finding that they have qualitatively similar forms, but differ quantitatively in the anticipated fashion.

Keywords

One-dimensional random walkstatistics of the maximumdiscrete probabilityasymptotic techniques

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 1 , March 2014 , pp. 162 - 173
Copyright
© Applied Probability Trust 

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