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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
33/2009

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

Andrea Collevecchio and Tom Schmitz

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 $b/(b+\delta)$, where $\delta$ is the reinforcement parameter. For $\delta >1$ we provide a lower bound of the form $\gamma^2 b/(b+\delta)$, where $\gamma$ is the survival probability of an associated branching process.

Received:
16.07.09
Published:
17.07.09
MSC Codes:
60K37, 60K99
Keywords:
random walk in a random environment, once edge-reinforced random walk, regeneration times

Related publications

inJournal
2011 Journal Open Access
Andrea Collevecchio and Tom Schmitz

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

In: The annals of applied probability, 21 (2011) 3, pp. 1073-1101