Argue that the Markov chain from Exercise 6b is not ergodic. Certainly, if this is true, Doeblin’s condition must not hold. Show directly that this is indeed the case. (Advice: Try to avoid long calculations.)

Exercise 6b

Show that the random walk process in Section 2.2.4 is a Markov chain. Find the transition matrices for the following two cases.

(b) The process comes to a stop when X_t = 0 or a, as it was assumed in Section 2.2.4. In this case, X_t may assume values 0,1, ...,a.