If both classes were transient, after some time, the chain would not be in either class.

If both classes were transient, after some time, the chain would not be in either class.

• A. False
• B. Cannot be determined
• C. Only if the chain is finite
• D. True

Answer: (D)

In a Markov chain, a communicating class is transient when the chain will leave it and, with probability 1, not return. States in a transient class are visited only finitely many times almost surely. If both classes in question are transient, starting from any state in those classes, the chain will eventually move away and not come back to either of them. Therefore, after some finite time, the chain would not be in either class. It could end up in some other class (possibly recurrent), but that does not contradict the statement about these two transient classes. This is why the statement is true, and it doesn’t depend on the chain being finite.

In a Markov chain, a communicating class is transient when the chain will leave it and, with probability 1, not return. States in a transient class are visited only finitely many times almost surely. If both classes in question are transient, starting from any state in those classes, the chain will eventually move away and not come back to either of them. Therefore, after some finite time, the chain would not be in either class. It could end up in some other class (possibly recurrent), but that does not contradict the statement about these two transient classes. This is why the statement is true, and it doesn’t depend on the chain being finite.