Spitzer's identity for discrete random walks
Probability
2017-10-27 v1 Combinatorics
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.
Cite
@article{arxiv.1710.09670,
title = {Spitzer's identity for discrete random walks},
author = {A. J. E. M. Janssen and Johan S. H. van Leeuwaarden},
journal= {arXiv preprint arXiv:1710.09670},
year = {2017}
}