The maximum of a branching random walk with semiexponential increments

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We consider an infinite Galton-Watson tree Γ and label the vertices v with a collection of i.i.d. random variables (Yv)v∈Γ. In the case where the upper tail of the distribution of Yv is semiexponential, we then determine the speed of the corresponding tree-indexed random walk. In contrast to the classical case where the random variables Yv have finite exponential moments, the normalization in the definition of the speed depends on the distribution of Yv. Interpreting the random variables Yv as displacements of the offspring from the parent, (Yv)v∈Γ describes a branching random walk. The result on the speed gives a limit theorem for the maximum of the branching random walk, that is, for the position of the rightmost particle. In our case, this maximum grows faster than linear in time.

Original languageEnglish
Pages (from-to)1219-1229
Number of pages11
JournalAnnals of Probability
Issue number3
StatePublished - Jul 2000
Externally publishedYes


  • Branching random walk
  • Galton-Watson tree
  • Semiexponential distributions
  • Sums of i.i.d. random variables
  • Tree-indexed random walk


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