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.