## Institute of Mathematical Statistics Lecture Notes - Monograph Series

- Lecture Notes--Monograph Series
- Number 50, 2006, 31-43

### Random walk on a polygon

#### Abstract

A particle moves among the vertices of an $(m+1)$-gon which are labeled clockwise as $0, 1, \ldots, m$. The particle starts at $0$ and thereafter at each step it moves to the adjacent vertex, going clockwise with a known probability $p$, or counterclockwise with probability $1-p$. The directions of successive movements are independent. What is the expected number of moves needed to visit all vertices? This and other related questions are answered using recursive relations.

#### Chapter information

**Source***Recent Developments in Nonparametric Inference and Probability: Festschrift for Michael Woodroofe* (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2006)

**Dates**

First available in Project Euclid: 28 November 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.lnms/1196284051

**Digital Object Identifier**

doi:10.1214/074921706000000581

**Subjects**

Primary: 60G50: Sums of independent random variables; random walks

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

**Keywords**

Gambler's ruin recursive relation conditional probability mathematical induction difference equation L'Hospital's rule Bayes' rule

**Rights**

Copyright © 2006, Institute of Mathematical Statistics

#### Citation

Sarka, Jyotirmoy. Random walk on a polygon. Recent Developments in Nonparametric Inference and Probability, 31--43, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2006. doi:10.1214/074921706000000581. https://projecteuclid.org/euclid.lnms/1196284051