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 language | English (US) |
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Pages (from-to) | 280-338 |
Number of pages | 59 |
Journal | Annals of Probability |
Volume | 40 |
Issue number | 1 |
DOIs | |
State | Published - 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