Show that the stationary probabilities for the Markov chain having transition probabilities Pi,j are also the stationary probabilities for the Markov chain whose transition probabilities Qi,j are...

Show that the stationary probabilities for the Markov chain having transition probabilities Pi,j are also the stationary probabilities for the Markov chain whose transition probabilities Qi,j are given by

for any specified positive integer k.

q159:

Recall that state i is said to be positive recurrent if mi,i <>

it follows that state i is positive recurrent if and only if πi > 0. Suppose that state i is positive recurrent and that state i communicates with state j. Show that state j is also positive recurrent by arguing that there is an integer n such that