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March 2014 The maximum of a symmetric next neighbor walk on the nonnegative integers
Ora E. Percus, Jerome K. Percus
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J. Appl. Probab. 51(1): 162-173 (March 2014). DOI: 10.1239/jap/1395771421

Abstract

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≤jnSn = 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≤jnSj = 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.

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Ora E. Percus. Jerome K. Percus. "The maximum of a symmetric next neighbor walk on the nonnegative integers." J. Appl. Probab. 51 (1) 162 - 173, March 2014. https://doi.org/10.1239/jap/1395771421

Information

Published: March 2014
First available in Project Euclid: 25 March 2014

zbMATH: 1306.60046
MathSciNet: MR3189449
Digital Object Identifier: 10.1239/jap/1395771421

Subjects:
Primary: 60G50
Secondary: 60J10

Rights: Copyright © 2014 Applied Probability Trust

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Vol.51 • No. 1 • March 2014
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