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A Delayed Yule Process


Authors: Radu Dascaliuc, Nicholas Michalowski, Enrique Thomann and Edward C. Waymire
Journal: Proc. Amer. Math. Soc. 146 (2018), 1335-1346
MSC (2010): Primary 60G05, 60G44; Secondary 35S35
DOI: https://doi.org/10.1090/proc/13905
Published electronically: October 30, 2017
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Abstract: In now classic work, David Kendall (1966) recognized that the Yule process and Poisson process could be related by a (random) time change. Furthermore, he showed that the Yule population size rescaled by its mean has an almost sure exponentially distributed limit as $ t\to \infty $. In this note we introduce a class of coupled delayed continuous time Yule processes parameterized by $ 0 < \alpha \le 1$ and find a representation of the Poisson process as a delayed Yule process at delay rate $ \alpha = {1/2}$. Moreover we extend Kendall's limit theorem to include a larger class of positive martingales derived from functionals that gauge the population genealogy. Specifically, the latter is exploited to uniquely characterize the moment generating functions of distributions of the limit martingales, generalizing Kendall's mean one exponential limit. A connection with fixed points of the Holley-Liggett smoothing transformation also emerges in this context, about which much is known from general theory in terms of moments, tail decay, and so on.


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Additional Information

Radu Dascaliuc
Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon, 97331
Email: dascalir@math.oregonstate.edu

Nicholas Michalowski
Affiliation: Department of Mathematics, New Mexico State University, Las Cruces, New Mexico, 88003

Enrique Thomann
Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon, 97331

Edward C. Waymire
Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon, 97331
Email: waymire@math.oregonstate.edu

DOI: https://doi.org/10.1090/proc/13905
Received by editor(s): July 25, 2016
Received by editor(s) in revised form: January 11, 2017, and May 9, 2017
Published electronically: October 30, 2017
Communicated by: David Levin
Article copyright: © Copyright 2017 American Mathematical Society

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