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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

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