The Annals of Probability

On explosions in heavy-tailed branching random walks

Omid Amini, Luc Devroye, Simon Griffiths, and Neil Olver

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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 $\lim_{n\rightarrow\infty}M_{n}=\infty$).

Article information

Source
Ann. Probab., Volume 41, Number 3B (2013), 1864-1899.

Dates
First available in Project Euclid: 15 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1368637061

Digital Object Identifier
doi:10.1214/12-AOP806

Mathematical Reviews number (MathSciNet)
MR3098061

Zentralblatt MATH identifier
1304.60093

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60F20: Zero-one laws
Secondary: 60C05: Combinatorial probability

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

Citation

Amini, Omid; Devroye, Luc; Griffiths, Simon; Olver, Neil. On explosions in heavy-tailed branching random walks. Ann. Probab. 41 (2013), no. 3B, 1864--1899. doi:10.1214/12-AOP806. https://projecteuclid.org/euclid.aop/1368637061


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