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

Ora E. Percus, Jerome K. Percus

Research output: Contribution to journalArticlepeer-review


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
Issue number1
StatePublished - Mar 2014


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

ASJC Scopus subject areas

  • Statistics and Probability
  • General Mathematics
  • Statistics, Probability and Uncertainty


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