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+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 γ2b/(b+δ), where
γ is the survival probability of an associated branching process.Comment: 21 page