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November 2017 A leader-election procedure using records
Gerold Alsmeyer, Zakhar Kabluchko, Alexander Marynych
Ann. Probab. 45(6B): 4348-4388 (November 2017). DOI: 10.1214/16-AOP1167

Abstract

Motivated by the open problem of finding the asymptotic distributional behavior of the number of collisions in a Poisson–Dirichlet coalescent, the following version of a stochastic leader-election algorithm is studied. Consider an infinite family of persons, labeled by $1,2,3,\ldots$, who generate i.i.d. random numbers from an arbitrary continuous distribution. Those persons who have generated a record value, that is, a value larger than the values of all previous persons, stay in the game, all others must leave. The remaining persons are relabeled by $1,2,3,\ldots$ maintaining their order in the first round, and the election procedure is repeated independently from the past and indefinitely. We prove limit theorems for a number of relevant functionals for this procedure, notably the number of rounds $T(M)$ until all persons among $1,\ldots,M$, except the first one, have left (as $M\to\infty$). For example, we show that the sequence $(T(M)-\log^{*}M)_{M\in \mathbb{N}}$, where $\log^{*}$ denotes the iterated logarithm, is tight, and study its weak subsequential limits. We further provide an appropriate and apparently new kind of normalization (based on tetrations) such that the original labels of persons who stay in the game until round $n$ converge (as $n\to\infty$) to some random non-Poissonian point process and study its properties. The results are applied to describe all subsequential distributional limits for the number of collisions in the Poisson–Dirichlet coalescent, thus providing a complete answer to the open problem mentioned above.

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Gerold Alsmeyer. Zakhar Kabluchko. Alexander Marynych. "A leader-election procedure using records." Ann. Probab. 45 (6B) 4348 - 4388, November 2017. https://doi.org/10.1214/16-AOP1167

Information

Received: 1 February 2016; Revised: 1 November 2016; Published: November 2017
First available in Project Euclid: 12 December 2017

zbMATH: 06838122
MathSciNet: MR3737913
Digital Object Identifier: 10.1214/16-AOP1167

Subjects:
Primary: 60F05, 60G55
Secondary: 60J10

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 6B • November 2017
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