Exercise 10.8 Let {Xn} be a homogeneous Markov chain defined on a finite state space N with transition matrix P. From (10.9), the state distribution

(n) is given by ⊤(n) = ⊤Pn. Show that the limiting distribution

 = limn→∞ (n) exists, if P is primitive (see Kijima [1997] for the definition).

Also, using (10.10), show that the limiting distribution is a stationary distribution in the sense that it satisfies ⊤ = ⊤P. Calculate the stationary distribution for the Markov chain given in Exercise 10.6. Hint: By the Perron–
Frobenius theorem, we have Pn = 1⊤+Δn, n = 1, 2, . . . , where the spectral radius of Δ is strictly less than unity.