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The Necklace Process

Part of: Geometric probability and stochastic geometry Combinatorial probability Inference from stochastic processes

Published online by Cambridge University Press:  14 July 2016

Colin Mallows  and
Larry Shepp
Colin Mallows*
Affiliation:
Avaya Labs
Larry Shepp*
Affiliation:
Rutgers University
*
∗Postal address: Avaya Labs, 233 Mt. Airy Road, Basking Ridge, NJ 07920, USA. Email address: colinm@research.avayalabs.com
∗∗Postal address: Department of Statistics, 501 Hill Center, Busch Campus, Rutgers University, Piscataway, NJ 08854-8019, USA. Email address: shepp@stat.rutgers.edu
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Abstract

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Start with a necklace consisting of one white bead and one black bead, and add new beads one at a time by inserting each new bead between a randomly chosen adjacent pair of old beads, with the proviso that the new bead will be white if and only if both beads of the adjacent pair are black. Let Wn denote the number of white beads when the total number of beads is n. We show that EWn = n/3 and, with c2 = 2/45, that (Wn − n/3) / c√n is asymptotically standard normal. We find that, for all r ≥ 1 and n > 2r, the rth cumulant of the distribution of Wn is of the form nhr. We find the expected numbers of gaps of given length between white beads, and examine the asymptotics of the longest gaps.

Keywords

Special nonlinear recurrenceMarkov chainurn model

MSC classification

Secondary: 60C05: Combinatorial probability 60D05: Geometric probability and stochastic geometry 62M05: Markov processes
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 1 , March 2008 , pp. 271 - 278
Copyright
Copyright © Applied Probability Trust 2008