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Expected coalescence time for a nonuniform allocation process

Part of: Limit theorems Combinatorial probability Algorithms - Computer Science Markov processes

Published online by Cambridge University Press:  01 July 2016

John K. McSweeney  and
Boris G. Pittel
John K. McSweeney*
Affiliation:
The Ohio State University
Boris G. Pittel*
Affiliation:
The Ohio State University
*
∗ Postal address: Department of Mathematics, The Ohio State University, 231 W 18th Avenue, Columbus, OH 43210, USA.
∗ Postal address: Department of Mathematics, The Ohio State University, 231 W 18th Avenue, Columbus, OH 43210, USA.
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Abstract

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We study a process where balls are repeatedly thrown into n boxes independently according to some probability distribution p. We start with n balls, and at each step, all balls landing in the same box are fused into a single ball; the process terminates when there is only one ball left (coalescence). Let c := ∑jpj2, the collision probability of two fixed balls. We show that the expected coalescence time is asymptotically 2c−1, under two constraints on p that exclude a thin set of distributions p. One of the constraints is c = o(ln−2n). This ln−2n is shown to be a threshold value: for c = ω(ln−2n), there exists p with c(p) = c such that the expected coalescence time far exceeds c−1. Connections to coalescent processes in population biology and theoretical computer science are discussed.

Keywords

Coalescencemost recent common ancestorgeneralized Wright-Fisher modelrandom functionMarkov chainasymptotic behavior

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 60F10: Large deviations 60J05: Discrete-time Markov processes on general state spaces 68W40: Analysis of algorithms 92D25: Population dynamics (general)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 4 , December 2008 , pp. 1002 - 1032
Copyright
Copyright © Applied Probability Trust 2008 

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