Spitzer's identity for discrete random walks

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Abstract

Spitzer's identity describes the position of a reflected random walk over time in terms of a bivariate transform. Among its many applications in probability theory are congestion levels in queues and random walkers in physics. We present a derivation of Spitzer's identity for random walks with bounded jumps to the left, only using basic properties of analytic functions and contour integration. The main novelty is a reversed approach that recognizes a factored polynomial expression as the outcome of Cauchy's formula.

Original languageEnglish
Pages (from-to)168-172
Number of pages5
JournalOperations Research Letters
Volume46
Issue number2
DOIs
Publication statusPublished - 1 Mar 2018

Keywords

  • Complex analysis
  • Fluctuation theory
  • Spitzer's identity
  • Transform methods

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