The maximum of a branching random walk with stretched exponential tails

Piotr Dyszewski, Nina Gantert, Thomas Höfelsauer

Research output: Contribution to journalArticlepeer-review


We study a one-dimensional branching random walk in the case when the step size distribution has a stretched exponential tail, and, in particular, no finite exponential moments. The tail of the step size X decays as P[X > t] ∼ a exp{−λtr } for some constants a, λ > 0 where r ∈ (0, 1). We give a detailed description of the asymptotic behaviour of the position of the rightmost particle, proving almost sure limit theorems, convergence in law and a growth condition dichotomy. The limit theorems reveal interesting differences betweens the two regimes r ∈ (0, 2/3) and r ∈ (2/3, 1), with yet different limits in the boundary case r = 2/3.

Original languageEnglish
Pages (from-to)539-562
Number of pages24
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Issue number2
StatePublished - May 2023


  • Branching random walk
  • Extreme values
  • Limit theorems
  • Point processes
  • Stretched exponential random variables


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