Estimate of a transition matrix. Let X0; : : : ; Xn be a Markov chain with finite state space E, known initial distribution ˛ and unknown transition matrix …. For a; b 2 E, let L.2/.a; b/ D j¹1i n W Xi1 D a; Xi D bºj=n be the relative frequency of the letter pair

.a; b/ in the ‘random word’ .X0; : : : ; Xn/. The random matrix L.2/ D .L.2/.a; b//a;b2E is called the empirical pair distribution. Define the empirical transition matrix T on E by T.a; b/ D L.2/.a; b/=L.a/ if L.a/ WD X

c2E L.2/.a; c/ > 0 ;

and arbitrarily otherwise. Specify the statistical model and show that T is a maximum likelihood estimator of …. Hint: You can argue as in Example (7.7).