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An Application of the Coalescence Theory to Branching Random Walks

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  30 January 2018

K. B. Athreya  and
Jyy-I Hong
K. B. Athreya*
Affiliation:
Iowa State University
Jyy-I Hong*
Affiliation:
Waldorf College
*
∗ Postal address: Iowa State University, Ames, Iowa 50011, USA.
∗∗ Postal address: Department of Mathematics, Waldorf College, 106 South Sixth Street, Forest City, IA 50436, USA. Email address: hongjyyi@gmail.com
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Abstract

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In a discrete-time single-type Galton--Watson branching random walk {Zn, ζn}n≤ 0, where Zn is the population of the nth generation and ζn is a collection of the positions on ℝ of the Zn individuals in the nth generation, let Yn be the position of a randomly chosen individual from the nth generation and Zn(x) be the number of points in ζn that are less than or equal to x for x∈ℝ. In this paper we show in the explosive case (i.e. m=E(Z1∣ Z0=1)=∞) when the offspring distribution is in the domain of attraction of a stable law of order α,0 <α<1, that the sequence of random functions {Zn(x)/Zn:−∞<x<∞} converges in the finite-dimensional sense to {δx:−∞<x<∞}, where δx≡ 1{N≤ x} and N is an N(0,1) random variable.

Keywords

Branching processbranching random walkcoalescencesupercriticalinfinite mean

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G50: Sums of independent random variables; random walks
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 3 , September 2013 , pp. 893 - 899
Copyright
© Applied Probability Trust 

References

Asmussen, S. and Kaplan, N. (1976). Branching random walks. I. Stoch. Process. Appl. 4, 1–13.CrossRefGoogle Scholar
Athreya, K. B. (2010). Branching random walks. In The Legacy of Alladi Ramakrishnan in the Mathematical Sciences, Springer, New York, pp. 337–349.CrossRefGoogle Scholar
Athreya, K. B. (2012). Coalescence in the recent past in rapidly growing populations. Stoch. Process. Appl. 122, 3757–3766.CrossRefGoogle Scholar
Biggins, J. D. (1990). The central limit theorem for the supercritical branching random walk, and related results. Stoch. Process. Appl. 34, 255–274.CrossRefGoogle Scholar
Davies, P. L. (1978). The simple branching process: a note on convergence when the mean is infinite. J. Appl. Prob. 15, 466–480.CrossRefGoogle Scholar
Grey, D. R. (1977). Almost sure convergence in Markov branching processes with infinite mean. J. Appl. Prob. 14, 702–716.CrossRefGoogle Scholar
Grey, D. R. (1979). On regular branching processes with infinite mean. Stoch. Process. Appl. 8, 257–267.CrossRefGoogle Scholar
Kaplan, N. and Asmussen, S. (1976). Branching random walks. II. Stoch. Process. Appl. 4, 15–31.CrossRefGoogle Scholar