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An equivalent class C of communicating states, called a communicating class, is said to be closed...

An equivalent class C of communicating states, called a communicating class, is said to be closed if Pi,j = 0 whenever i ∈ C and j 6∈ C. In other words, a communicating class is closed if there is no escape from that class.

(a) Show that every recurrent class is closed.

(b) Show that every Markov chain on a finite state space has at least one closed communicating class.

(c) Find an example of a Markov chain with no closed communicating class.

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