ON EXPLOSIONS IN HEAVY-TAILED BRANCHING RANDOM WALKS

• Published in: The Annals of probability 2013-05, Vol.41 (3), p.1864-1899
• Main Authors: Amini, Omid, Devroye, Luc, Griffiths, Simon, Olver, Neil
• Format: Article
• Language: English
• Subjects: 60C05; 60F20; 60J80; Atoms; Atoms & subatomic particles; Branching random walk; Children; Equivalence relation; explosion; Explosions; Galton–Watson trees; Generating function; Infinity; Integers; min-summability; Plant roots; Probability distribution; Random variables; Random walk; Random walk theory; speed of a Galton–Watson process; Studies; Sufficient conditions
• ISSN: 0091-1798; 2168-894X
• DOI: 10.1214/12-AOP806

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 M n , the position of the particle in generation n closest to the origin, when explosion does not occur (and hence lim n→∞ M n = ∞).