MiS Preprint Repository

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.