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January 2012 Biased random walks on Galton–Watson trees with leaves
Gérard Ben Arous, Alexander Fribergh, Nina Gantert, Alan Hammond
Ann. Probab. 40(1): 280-338 (January 2012). DOI: 10.1214/10-AOP620


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.


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Gérard Ben Arous. Alexander Fribergh. Nina Gantert. Alan Hammond. "Biased random walks on Galton–Watson trees with leaves." Ann. Probab. 40 (1) 280 - 338, January 2012.


Published: January 2012
First available in Project Euclid: 3 January 2012

zbMATH: 1239.60091
MathSciNet: MR2917774
Digital Object Identifier: 10.1214/10-AOP620

Primary: 60F05 , 60J80 , 60K37
Secondary: 60E07

Keywords: Electrical networks , Galton–Watson tree , Infinitely divisible distributions , Random walk in random environment

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 1 • January 2012
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