Journal of Applied Probability

An application of the coalescence theory to branching random walks

K. B. Athreya and Jyy-I Hong

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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(Z1Z0=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 δx1{Nx} and N is an N(0,1) random variable.

Article information

J. Appl. Probab., Volume 50, Number 3 (2013), 893-899.

First available in Project Euclid: 5 September 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60G50: Sums of independent random variables; random walks

Branching process branching random walk coalescence supercritical infinite mean


Athreya, K. B.; Hong, Jyy-I. An application of the coalescence theory to branching random walks. J. Appl. Probab. 50 (2013), no. 3, 893--899. doi:10.1239/jap/1378401245.

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