I’ve asked David Steinsaltz about problem 5 (b) in sheet 3, and he tells me that he stated a theorem (13.7) last week that allows you to compute long term averages using stationary states even in the non-ergodic case.
Recall that the problem is to compute
The probability can be written as , where and is the transition
matrix. The theorem says that this limit is the same as the average distance for the stationary distribution, which is 3/2.
One way to reduce this problem to the ergodic case is to note that the processes that act as on the odd distance states and the even distance state separately are both ergodic. If you compute the stationary states in these cases, you will note that the average distance for both stationary distributions is 3/2. Hence, the limit above will also converge to 3/2.