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Power Law Condition for Stability of Poisson Hail

Author

Listed:
  • Sergey Foss

    (Heriot-Watt University
    Sobolev Institute of Mathematics and Novosibirsk State University)

  • Takis Konstantopoulos

    (Uppsala University)

  • Thomas Mountford

    (Ecole Polytechnique Fédérale de Lausanne)

Abstract

The Poisson hail model is a space-time stochastic system introduced by Baccelli and Foss (J Appl Prob 48A:343–366, 2011) whose stability condition is nonobvious owing to the fact that it is spatially infinite. Hailstones arrive at random points of time and are placed in random positions of space. Upon arrival, if not prevented by previously accumulated stones, a stone starts melting at unit rate. When the stone sizes have exponential tails, then stability conditions exist. In this paper, we look at heavy tailed stone sizes and prove that the system can be stabilized when the rate of arrivals is sufficiently small. We also show that the stability condition is, in a weak sense, optimal. We use techniques and ideas from greedy lattice animals.

Suggested Citation

  • Sergey Foss & Takis Konstantopoulos & Thomas Mountford, 2018. "Power Law Condition for Stability of Poisson Hail," Journal of Theoretical Probability, Springer, vol. 31(2), pages 684-704, June.
    • Handle: RePEc:spr:jotpro:v:31:y:2018:i:2:d:10.1007_s10959-016-0723-3
    DOI: 10.1007/s10959-016-0723-3
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    References listed on IDEAS

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    1. Martin, James B., 2002. "Linear growth for greedy lattice animals," Stochastic Processes and their Applications, Elsevier, vol. 98(1), pages 43-66, March.
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