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On explosions in heavy-tailed branching random walks

Amini, Omid, Devroye, Luc, Griffiths, Simon and Olver, Neil (2013) On explosions in heavy-tailed branching random walks. Annals of Probability, 41 (3B). 1864 - 1899. ISSN 1864-1899

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Identification Number: 10.1214/12-AOP806

Abstract

Consider a branching random walk on 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 Mn, the position of the particle in generation n closest to the origin, when explosion does not occur (and hence limn→∞Mn=∞).

Item Type: Article
Official URL: https://projecteuclid.org/all/euclid.aop
Additional Information: © 2013 Institute of Mathematical Statistics
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 20 Jan 2020 10:03
Last Modified: 20 May 2020 04:11
URI: http://eprints.lse.ac.uk/id/eprint/103074

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