Biased random walks on Galton–Watson trees with leaves

• DOI: 10.1214/10-aop620
• Publication Date: 2012-01-03T15:57:42Z
• Authors: Gérard Ben Arous; Alexander Fribergh; Nina Gantert; Alan Hammond
• Keywords: Tree (set theory); Constant (computer programming)

We consider a biased random walk Xn on Galton–Watson tree with leaves in the sub-ballistic regime. prove that there exists an explicit constant γ = γ(β) ∈ (0, 1), depending bias β, such |Xn| is of order nγ. Denoting Δn hitting time level n, we Δn/n1/γ tight. Moreover, show does not converge law (at least for large values β). along sequences nλ(k) ⌊λβγk⌋, converges to certain infinitely divisible laws. Key tools proof are classical Harris decomposition trees, new variant regeneration times and careful analysis triangular arrays i.i.d. heavy-tailed variables.