### 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 language | English |
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Pages (from-to) | 168-172 |

Number of pages | 5 |

Journal | Operations Research Letters |

Volume | 46 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Mar 2018 |

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### Keywords

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

### Cite this

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*Operations Research Letters*, vol. 46, no. 2, pp. 168-172. https://doi.org/10.1016/j.orl.2017.12.003

**Spitzer's identity for discrete random walks.** / Janssen, A.J.E.M.; van Leeuwaarden, J. S.H.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Spitzer's identity for discrete random walks

AU - Janssen, A.J.E.M.

AU - van Leeuwaarden, J. S.H.

PY - 2018/3/1

Y1 - 2018/3/1

N2 - 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.

AB - 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.

KW - Complex analysis

KW - Fluctuation theory

KW - Spitzer's identity

KW - Transform methods

UR - http://www.scopus.com/inward/record.url?scp=85040306793&partnerID=8YFLogxK

U2 - 10.1016/j.orl.2017.12.003

DO - 10.1016/j.orl.2017.12.003

M3 - Article

AN - SCOPUS:85040306793

VL - 46

SP - 168

EP - 172

JO - Operations Research Letters

JF - Operations Research Letters

SN - 0167-6377

IS - 2

ER -