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

Ora E. Percus, Jerome Percus

Research output: Contribution to journalArticle

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 ¥[Sn = x, maxi<7< Sn = 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 maxi<7< Sj = 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.

Original languageEnglish (US)
Pages (from-to)162-173
Number of pages12
JournalJournal of Applied Probability
Volume51
Issue number1
DOIs
StatePublished - 2014

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Walk
Non-negative
Integer
Generating Function
Random walk
Probability Distribution
Probability distribution
Generating function
Form

Keywords

  • Asymptotic techniques
  • Discrete probability
  • One-dimensional random walk
  • Statistics of the maximum

ASJC Scopus subject areas

  • Mathematics(all)
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

The maximum of a symmetric next neighbor walk on the nonnegative integers. / Percus, Ora E.; Percus, Jerome.

In: Journal of Applied Probability, Vol. 51, No. 1, 2014, p. 162-173.

Research output: Contribution to journalArticle

Percus, Ora E. ; Percus, Jerome. / The maximum of a symmetric next neighbor walk on the nonnegative integers. In: Journal of Applied Probability. 2014 ; Vol. 51, No. 1. pp. 162-173.
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