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 tran- sient random walk in a random environment on regular trees. A refinement of this technique yields upper bounds on the first regeneration level and re- generation 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 + δ) for the speed on the b-ary tree, where δ is the reinforcement parameter. For δ > 1 we provide a lower bound of the form γ 2b/(b + δ), where γ is the survival probability of an associated branching process.