Exercise 10.11 In Equation (10.15), suppose that the underlying Markov chain is finite and homogeneous, that is, Q(t, t + 1) = Q for all t. Show that the quasi-limiting distribution ˆ = limn→∞ ˆ(n) exists, if Q is primitive. Also, using (10.14), show that the quasi-limiting distribution is a quasi-stationary distribution in the sense that it satisfies ˆ

⊤

= γ ˆ

⊤

Q for some γ > 0. Calculate the quasi-stationary distribution for the Markov chain given in Exercise 10.10.

Hint: Use the Perron–Frobenius theorem.