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Abstract: We study the extremes of branching random walks under the assumption that underlying Galton-Watson tree has infinite progeny mean. It is assumed that the displacements are either regularly varying or 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. In the lighter-tailed case, however, the behaviour is much more subtle, and the scaling of the position of the rightmost particle in the n-th generation depends on the family of stepsize distribution, not just its parameter(s). In all of these cases, we discuss the convergence in probability of the scaled maxima sequence. Our results and methodology are applied to study the almost sure convergence in the context of cloud speed for branching random walks with infinite progeny mean. The exact cloud speed constants are calculated for regularly varying displacements and also for stepsize distributions having a nice exponential decay.
This talk is based on a joint work with Souvik Ray (Stanford University), Rajat Subhra Hazra (ISI Kolkata) and Philippe Soulier (Univ of Paris Nanterre). We will first review the literature (mainly, the PhD thesis work of Ayan Bhattacharya) and then talk about the current work. Special care will be taken so that a significant portion of the talk remains accessible to everyone.
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