On explosions in heavy-tailed branching random walks

• Omid Amini ^[1] ; Luc Devroye ^[2] ; Simon Griffiths ^[3] ; Neil Olver ^[4]
  1. [1] École Normale Supérieure

     École Normale Supérieure

     Francia

     
  2. [2] McGill University

     McGill University

     Canadá

     
  3. [3] Instituto Nacional de Matemática Pura e Aplicada

     Instituto Nacional de Matemática Pura e Aplicada

     Brasil

     
  4. [4] MIT

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• Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 41, Nº. 3, 2, 2013, págs. 1864-1899
• Idioma: inglés
• DOI: 10.1214/12-AOP806
• Texto completo no disponible (Saber más ...)
• Resumen

  • 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=∞).