Biased random walks on Galton-Watson trees with leaves

Gérard Ben Arous, Alexander Fribergh, Nina Gantert, Alan Hammond

Research output: Contribution to journalArticle

Abstract

We consider a biased random walk X n on a Galton-Watson tree with leaves in the sub-ballistic regime. We prove that there exists an explicit constant γ = γ(β) ∈ (0, 1), depending on the bias β, such that |X n| is of order n γ. Denoting {increment} n the hitting time of level n, we prove that {increment} n/n 1/γ is tight. Moreover, we show that {increment} n/n 1/γ does not converge in law (at least for large values of β). We prove that along the sequences n λ(k) =⌊λβ γk⌋, {increment} n/n 1/γ converges to certain infinitely divisible laws. Key tools for the proof are the classical Harris decomposition for Galton-Watson trees, a new variant of regeneration times and the careful analysis of triangular arrays of i.i.d. heavy-tailed random variables.

Original languageEnglish (US)
Pages (from-to)280-338
Number of pages59
JournalAnnals of Probability
Volume40
Issue number1
DOIs
StatePublished - Jan 2012

Keywords

  • Electrical networks
  • Galton-Watson tree
  • Infinitely divisible distributions
  • Random walk in random environment

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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