Is the following statement always true? “If an irreducible Markov chain has period 2, then for...

Is the following statement always true? “If an irreducible Markov chain has period 2, then for every state i∈S we have P2ii>0.” (Prove if “yes”, provide a counterexample if “no”)

Solutions

Expert Solution

The period of a state is defined by

In our case, we have

Consider the Markov chain with the following transition matrix

Its transition diagram is the following:

Notice that all states are strongly connected to each other, making the chain irreducible. Moreover, all state has period . This is obvious for all states other than . For state , period is

Notice that because there is no length- path in the transition diagram, from state to itself. This counterexample shows that the statement in question is not true.

Colby Messinger answered 1 year ago

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