Bounds on the speed and on regeneration times for certain processes on regular trees

Andrea Collevecchio and Tom Schmitz

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Submission date: 16. Jul. 2009
Pages: 23
published in: The annals of applied probability, 21 (2011) 3, p. 1073-1101 
DOI number (of the published article): 10.1214/10-AAP719
Bibtex
MSC-Numbers: 60K37, 60K99
Keywords and phrases: random walk in a random environment, once edge-reinforced random walk, regeneration times
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Abstract:
We develop a technique that provides a lower bound on the speed of transient random walk in a random environment on regular trees. A refinement of this technique yields upper bounds on the first regeneration level and regeneration time. In particular, a lower and upper bound on the covariance in the annealed invariance principle follows. Our methods are general and also apply in the case of once edge-reinforced random walk. Durrett, Kesten and Limic (Probab. Theory and Relat. Fields (122), 2002, p.567-592) prove an upper bound of the form , where is the reinforcement parameter. For we provide a lower bound of the form , where is the survival probability of an associated branching process.