In a 5-state Markov chain with two classes {0,1,3} and {2,4}, at least one of the two classes must be recurrent.

In a 5-state Markov chain with two classes {0,1,3} and {2,4}, at least one of the two classes must be recurrent.

• A. False
• B. Cannot be determined
• C. True
• D. Only if both classes are recurrent

Answer: (C)

In a finite Markov chain, states group into communicating classes, and a class is recurrent exactly when it is closed (no transitions leave the class). There must be at least one closed class in a finite chain, because if every class could be left, you’d have a way to move indefinitely without settling, which cannot happen with finitely many states. Here the chain has two classes, and there are no other states outside them. If both classes were non-closed, each would have a path to the other. That would mean the two classes actually communicate with each other and should form a single communicating class, contradicting the given two-class partition. Hence at least one class must be closed, i.e., recurrent. This makes the statement true.

In a finite Markov chain, states group into communicating classes, and a class is recurrent exactly when it is closed (no transitions leave the class). There must be at least one closed class in a finite chain, because if every class could be left, you’d have a way to move indefinitely without settling, which cannot happen with finitely many states.

Here the chain has two classes, and there are no other states outside them. If both classes were non-closed, each would have a path to the other. That would mean the two classes actually communicate with each other and should form a single communicating class, contradicting the given two-class partition. Hence at least one class must be closed, i.e., recurrent. This makes the statement true.