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May 2013 On explosions in heavy-tailed branching random walks
Omid Amini, Luc Devroye, Simon Griffiths, Neil Olver
Ann. Probab. 41(3B): 1864-1899 (May 2013). DOI: 10.1214/12-AOP806

Abstract

Consider a branching random walk on $\mathbb{R}$, with offspring distribution $Z$ and nonnegative displacement distribution $W$. We say that explosion occurs if an infinite number of particles may be found within a finite distance of the origin. In this paper, we investigate this phenomenon when the offspring distribution $Z$ is heavy-tailed. Under an appropriate condition, we are able to characterize the pairs $(Z,W)$ for which explosion occurs, by demonstrating the equivalence of explosion with a seemingly much weaker event: that the sum over generations of the minimum displacement in each generation is finite. Furthermore, we demonstrate that our condition on the tail is best possible for this equivalence to occur.

We also investigate, under additional smoothness assumptions, the behavior of $M_{n}$, the position of the particle in generation $n$ closest to the origin, when explosion does not occur (and hence $\lim_{n\rightarrow\infty}M_{n}=\infty$).

Citation

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Omid Amini. Luc Devroye. Simon Griffiths. Neil Olver. "On explosions in heavy-tailed branching random walks." Ann. Probab. 41 (3B) 1864 - 1899, May 2013. https://doi.org/10.1214/12-AOP806

Information

Published: May 2013
First available in Project Euclid: 15 May 2013

zbMATH: 1304.60093
MathSciNet: MR3098061
Digital Object Identifier: 10.1214/12-AOP806

Subjects:
Primary: 60F20 , 60J80
Secondary: 60C05

Keywords: Branching random walk , explosion , Galton–Watson trees , min-summability , speed of a Galton–Watson process

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 3B • May 2013
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