Let,,j=0, be the stationary probabilities for a specified Markov chain.

(a) Complete the following statement: 7,P., is the proportion of all transitions that.... Let A denote a set of states and let A' denote the remaining states.

(b) Finish the following statement: P, is the proportion of all transitions that .... JEAEA

(c) Let N,(A, A') denote the number of the first n transitions that are from a state in A to one in A'; similarly, let N,(A', A) denote the number that are from a state in A' to one in A. Argue that |N,(A, A) - N,(A', A)| 1.

(d) Prove and interpret the following result: - JEA IEA JEA LEA