Markov chain with infinite number of transient and positive recurrent states?

Question

Is it possible to have a markov chain with an infinite number of transient states, and an infinite number of positive recurrent states?

Thank you!

Answer 1

The chain must be reducible for this to happen. For an example, consider a nearest-neighbour random walk on the integer line, recurrent on the negative part, positive recurrent on the positive part, and with no transition from the latter to the former.

To be specific, consider that $p(x,x+1)=p_x$ and $p(x,x-1)=1-p_x$ for every $x$, $p_x=\frac12$ for every $x\leqslant-1$, $p_0=p_1=1$ and $p_x=\frac13$ for every $x\geqslant2$. Then every $x\leqslant0$ is transient and every $x\geqslant1$ is positive recurrent.