Journal of Applied Probability

The maximum of a symmetric next neighbor walk on the nonnegative integers

Ora E. Percus and Jerome K. Percus

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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≤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.

Article information

J. Appl. Probab., Volume 51, Number 1 (2014), 162-173.

First available in Project Euclid: 25 March 2014

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

One-dimensional random walk statistics of the maximum discrete probability asymptotic techniques


Percus, Ora E.; Percus, Jerome K. The maximum of a symmetric next neighbor walk on the nonnegative integers. J. Appl. Probab. 51 (2014), no. 1, 162--173. doi:10.1239/jap/1395771421.

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