# In Markov Chains, what is the difference between null recurrent and positive recurrent?

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?

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.