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

Fingerprint

Galton-Watson Tree
Increment
Biased
Random walk
Leaves
Infinitely Divisible Laws
Converge
Triangular Array
Hitting Time
Regeneration
Ballistics
Random variable
Decompose
Decomposition
Random variables

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

Cite this

Biased random walks on Galton-Watson trees with leaves. / Ben Arous, Gérard; Fribergh, Alexander; Gantert, Nina; Hammond, Alan.

In: Annals of Probability, Vol. 40, No. 1, 01.2012, p. 280-338.

Research output: Contribution to journalArticle

Ben Arous, G, Fribergh, A, Gantert, N & Hammond, A 2012, 'Biased random walks on Galton-Watson trees with leaves', Annals of Probability, vol. 40, no. 1, pp. 280-338. https://doi.org/10.1214/10-AOP620
Ben Arous, Gérard ; Fribergh, Alexander ; Gantert, Nina ; Hammond, Alan. / Biased random walks on Galton-Watson trees with leaves. In: Annals of Probability. 2012 ; Vol. 40, No. 1. pp. 280-338.
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