More Randomness of Environment Does Not Always Slow Down a Random Walk

We consider a random walk on Z in a stationary and ergodic random environment, whose states are called types of the vertices of Z . We find conditions for which the speed of the random walk is positive. In the case of a Markov chain environment with finitely many states, we give an explicit formula for the speed and for the asymptotic proportion of time spent at vertices of a certain type. Using these results, we compare the speed of random walks on Z in environments of varying randomness.