The Annals of Probability

Biased random walks on Galton–Watson trees with leaves

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

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Abstract

We consider a biased random walk Xn 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 |Xn| is of order nγ. Denoting Δn the hitting time of level n, we prove that Δn/n1/γ is tight. Moreover, we show that Δn/n1/γ does not converge in law (at least for large values of β). We prove that along the sequences nλ(k) = ⌊λβγk⌋, Δn/n1/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.

Article information

Source
Ann. Probab., Volume 40, Number 1 (2012), 280-338.

Dates
First available in Project Euclid: 3 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.aop/1325605004

Digital Object Identifier
doi:10.1214/10-AOP620

Mathematical Reviews number (MathSciNet)
MR2917774

Zentralblatt MATH identifier
1239.60091

Subjects
Primary: 60K37: Processes in random environments 60F05: Central limit and other weak theorems 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60E07: Infinitely divisible distributions; stable distributions

Keywords
Random walk in random environment Galton–Watson tree infinitely divisible distributions electrical networks

Citation

Ben Arous, Gérard; Fribergh, Alexander; Gantert, Nina; Hammond, Alan. Biased random walks on Galton–Watson trees with leaves. Ann. Probab. 40 (2012), no. 1, 280--338. doi:10.1214/10-AOP620. https://projecteuclid.org/euclid.aop/1325605004


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