Spitzer's identity for discrete random walks

TL;DR

This paper provides a new, simplified derivation of Spitzer's identity for discrete random walks with bounded jumps, using basic complex analysis techniques, broadening its accessibility and potential applications.

Contribution

A novel derivation of Spitzer's identity under bounded jumps, employing elementary analytic methods and contour integration, with a reversed approach based on Cauchy's formula.

Findings

Simplified proof of Spitzer's identity

Applicable to random walks with bounded jumps

Potential for broader application in queueing and physics

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.