Question 11.5.13: Let P be the transition matrix of an ergodic Markov chain an......
Let P be the transition matrix of an ergodic Markov chain and P∗ the reverse transition matrix. Show that they have the same fixed probability vector w.
Assume that w is a fixed vector for P. Then
\sum\limits_{i}{w_ip^*_{ij}} = \sum\limits_{i}{\frac{w_iw_jp_{ji}}{w_i} }= \sum\limits_{i}{w_jp_{ji}}=w_j,
so w is a fixed vector for P*. Thus if w* is the unique fixed vector for P* we must have w = w*.
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