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Necklace Processes Via Pólya Urns

Part of: Combinatorial probability Limit theorems

Published online by Cambridge University Press:  14 July 2016

Toshio Nakata
Toshio Nakata*
Affiliation:
Fukuoka University of Education
*
∗Postal address: Department of Mathematics, Fukuoka University of Education, Akama-Bunkyomachi, Munakata, Fukuoka, 811-4192, Japan. Email address: nakata@fukuoka-edu.ac.jp
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Abstract

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Mallows and Shepp (2008) developed the following necklace processes. Start with a necklace consisting of one white bead and one black bead, and insert, one at a time, under a deterministic rule, a white bead or a black bead between a randomly chosen adjacent pair. They studied the statistical properties of the number of white beads by investigating the nature of the moments and the expected number of gaps of given length between white beads. In this note we study the number of white beads via Pólya urns and give a classification of necklace processes for some general rules. Additionally, we discuss the number of runs, i.e. the number of consecutive same color beads, instead of the number of gaps.

Keywords

Necklace processPólya urnlaw of large numberscentral limit theorem

MSC classification

Secondary: 60C05: Combinatorial probability 60F05: Central limit and other weak theorems
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 1 , March 2009 , pp. 284 - 295
Copyright
Copyright © Applied Probability Trust 2009 

Footnotes

Dedicated to Professor Izumi Kubo on the occasion of his 70th birthday.

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