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

Article Type

Research Article

Publication Title

Stochastic Processes and their Applications

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.

First Page

120

Last Page

160

DOI

https://10.1016/j.spa.2023.03.001

Publication Date

6-1-2023

Comments

Open Access, Hybrid Gold, Green

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