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Let $P$ be the transition probability matrix for the homogeneous Markov chain on the states $E=\{1, 2, 3 \}$.

$$ P = \left[ \begin{array}{ccc} 2/3 & 1/3 & 0 \\ 1/3 & 1/3 & 1/3 \\ 0 & 0 & 1 \end{array} \right] $$

States $E=\{ 1 \}$ and $E=\{ 2 \}$ are transient (if I am not mistaken, because it is possible to not go back to them). Whereas state $E= \{ 3 \}$ is recurrent because it is possible to go back there.

Am I correct in thinking it is null-recurrent, because only state $E= \{ 3 \}$ can ever be returned to? Whereas hypothetically if another recurrent state existed, then they would both be positive-recurrent?

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In a Markov chain with finitely many states, all recurrent states are positive-recurrent. So in your example, the state $\{ 3 \}$ is positive-recurrent.

A null-recurrent state is a state the Markov chain will almost-surely return to, but the average of the return time will be infinite. So if you have a recurrent state $i$ in a finite Markov chain, there will only be finitely many states the chain can bounce between before it returns to $i$. Intuitively, the average time spent in those finite states, before the chain returns to $i$, cannot be infinite.

You can also check some of the related questions. Hope this helps!

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