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

Souvik Ray, Rajat Subhra Hazra, Parthanil Roy and Philippe Soulier

Stochastic Processes and their Applications, 2023, vol. 160, issue C, 120-160

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.

Keywords: Branching random walk; Galton–Watson tree with infinite progeny mean; Cloud speed; Point processes; Extremes (search for similar items in EconPapers)
Date: 2023
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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:160:y:2023:i:c:p:120-160

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DOI: 10.1016/j.spa.2023.03.001

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