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Branching random walk with infinite progeny mean: A tale of two tails

Author

Listed:
  • Ray, Souvik
  • Hazra, Rajat Subhra
  • Roy, Parthanil
  • Soulier, Philippe

Abstract

We study the extremes of branching random walks under the assumption that the underlying Galton–Watson tree has infinite progeny mean. It is assumed that the displacements are either regularly varying or they have lighter tails. In the regularly varying case, it is shown that the point process sequence of normalized extremes converges to a Poisson random measure. We study the asymptotics of the scaled position of the rightmost particle in the nth generation when the tail of the displacement behaves like exp(−K(x)), where either K is a regularly varying function of index r>0, or K has an exponential growth. We identify the exact scaling of the maxima in all cases and show the existence of a non-trivial limit when r>1.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:160:y:2023:i:c:p:120-160
    DOI: 10.1016/j.spa.2023.03.001
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    References listed on IDEAS

    as
    1. 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.
    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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