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Branching random walks, stable point processes and regular variation

Author

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  • Bhattacharya, Ayan
  • Hazra, Rajat Subhra
  • Roy, Parthanil

Abstract

Using the theory of regular variation, we give a sufficient condition for a point process to be in the superposition domain of attraction of a strictly stable point process. This sufficient condition is used to obtain the weak limit of a sequence of point processes induced by a branching random walk with jointly regularly varying displacements. Because of heavy tails of the step size distribution, we can invoke a one large jump principle at the level of point processes to give an explicit representation of the limiting point process. As a consequence, we extend the main result of Durrett (1983) and verify that two related predictions of Brunet and Derrida (2011) remain valid for this model.

Suggested Citation

  • Bhattacharya, Ayan & Hazra, Rajat Subhra & Roy, Parthanil, 2018. "Branching random walks, stable point processes and regular variation," Stochastic Processes and their Applications, Elsevier, vol. 128(1), pages 182-210.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:1:p:182-210
    DOI: 10.1016/j.spa.2017.04.009
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    References listed on IDEAS

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    1. Fasen, Vicky & Roy, Parthanil, 2016. "Stable random fields, point processes and large deviations," Stochastic Processes and their Applications, Elsevier, vol. 126(3), pages 832-856.
    2. Durrett, Richard, 1979. "Maxima of branching random walks vs. independent random walks," Stochastic Processes and their Applications, Elsevier, vol. 9(2), pages 117-135, November.
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    Cited by:

    1. Ekaterina Vl. Bulinskaya, 2021. "Maximum of Catalytic Branching Random Walk with Regularly Varying Tails," Journal of Theoretical Probability, Springer, vol. 34(1), pages 141-161, March.
    2. Ray, Souvik & Hazra, Rajat Subhra & Roy, Parthanil & Soulier, Philippe, 2023. "Branching random walk with infinite progeny mean: A tale of two tails," Stochastic Processes and their Applications, Elsevier, vol. 160(C), pages 120-160.

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