Bounds on the Speed and on Regeneration Times for Certain Processes on Regular Trees

Details Bounds on the Speed and on Regeneration Times for Certain Processes on Regular Trees Bounds on the Speed and on Regeneration Times for Certain Processes on Regular Trees

2009-11-02

arxiv.org/abs/0911.0305v1

Mathematics

Probability

21 pages

Scientific paper

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. We emphasize the fact that our methods are general and also apply in the case of once-reinforced random walk. Durrett, Kesten and Limic (2002) prove an upper bound of the form $b/(b+\delta)$ for the speed on the $b$-ary tree, 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.

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