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

TL;DR

This paper introduces new bounds on the speed and regeneration times of certain stochastic processes on regular trees, using a versatile technique applicable to various models including reinforced random walks.

Contribution

It develops a general method to bound the speed and regeneration times for processes on regular trees, extending previous results to reinforced random walks and providing new lower bounds.

Findings

Established lower bounds on the speed of random walks in random environments.

Derived upper bounds on regeneration levels and times.

Extended bounds to reinforced random walks with reinforcement parameter elta>1.

Abstract

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.