An Application of the Coalescence Theory to Branching Random Walks

In a discrete-time single-type Galton--Watson branching random walk {Z n , ζ n } n≤ 0, where Z n is the population of the nth generation and ζ n is a collection of the positions on ℝ of the Z n individuals in the nth generation, let Y n be the position of a randomly chosen individual from the nth ge...

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Bibliographic Details
Published in:Journal of applied probability 2013-09, Vol.50 (3), p.893-899
Main Authors: Athreya, K. B., Hong, Jyy-I
Format: Article
Language:English
Subjects:
60G50
60J80
Attraction
Branching process
branching random walk
Central limit theorem
coalescence
Coalescing
Collection
Cumulative distribution functions
Explosions
Explosives
infinite mean
Law
Mathematical functions
Mathematical theorems
Perceptron convergence procedure
Probability distribution
Random variables
Random walk
Random walk theory
Short Communications
Studies
supercritical
Symbols
Theoretical mathematics
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Summary:In a discrete-time single-type Galton--Watson branching random walk {Z n , ζ n } n≤ 0, where Z n is the population of the nth generation and ζ n is a collection of the positions on ℝ of the Z n individuals in the nth generation, let Y n be the position of a randomly chosen individual from the nth generation and Z n (x) be the number of points in ζ n that are less than or equal to x for x∈ℝ. In this paper we show in the explosive case (i.e. m=E(Z 1∣ Z 0=1)=∞) when the offspring distribution is in the domain of attraction of a stable law of order α,0
ISSN:0021-9002
1475-6072
DOI:10.1239/jap/1378401245