Athreya, K. B. ; Hong, Jyy-I
(2013)
*An Application of the Coalescence Theory to Branching Random Walks*
Journal of Applied Probability, 50
(3).
pp. 893-899.
ISSN 0021-9002

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Official URL: http://doi.org/10.1239/jap/1378401245

Related URL: http://dx.doi.org/10.1239/jap/1378401245

## Abstract

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.

Item Type: | Article |
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Source: | Copyright of this article belongs to Applied Probability Trust |

Keywords: | Branching process, branching random walk, coalescence, supercritical, infinite mean |

ID Code: | 131582 |

Deposited On: | 07 Dec 2022 07:46 |

Last Modified: | 07 Dec 2022 07:46 |

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