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

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 [Probab. Theory Related Fields. 122 (2002) 567.592] prove an upper bound of the form b/(b+.) for the speed on the b-ary tree, where . is the reinforcement parameter. For .>1 we provide a lower bound of the form .2b/(b+.), where . is the survival probability of an associated branching process.