Spitzer's identity for discrete random walks

DOI10.1016/J.ORL.2017.12.003zbMath1525.60091arXiv1710.09670MaRDI QIDQ6293072

Augustus J. E. M. Janssen, Johan S. H. van Leeuwaarden

Publication date: 26 October 2017

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 new derivation of Spitzer's identity under the assumption that the increments of the random walk have bounded jumps to the left. This mild assumption facilitates a proof of Spitzer's identity that only uses basic properties of analytic functions and contour integration. The main novelty, believed to be of broader interest, is a reversed approach that recognizes a factored polynomial expression as the outcome of Cauchy's formula.

Mathematics Subject Classification ID

Queueing theory (aspects of probability theory) (60K25) Markov chains (discrete-time Markov processes on discrete state spaces) (60J10)

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