Example of a markov chain with transient and recurrent states

Question

As the title says, I can't come up with an example of a markov chain with all possible states (transient, positive recurrent and null recurrent). I know that the state space must be infinite, otherwise all recurrent classes are positive recurrent. Any suggestions?

Answer 1

Consider a state space $\mathcal S=\{0\}\cup \{\delta\}\cup A\cup B$, where $A=\{a_1,a_2,\ldots\}$ and $B=\{b_1,b_2,\ldots\}$, and transition probabilities $$ P_{ij} = \begin{cases} \frac13,& i=0, j\in\{\delta, a_1,b_1\}\\ 1,& i = j = \delta\\ \frac13,& i=j=a_1\\ \frac23,& i=a_n, j=a_{n+1}\\ \frac13,& i=a_{n+1}, j=a_n\\ \frac23,& i=j=b_1\\ \frac13,& i=b_n, j=b_{n+1}\\ \frac23,& i=b_{n+1}, j=b_n.\\ \end{cases} $$ Then $0$ is transient, $\delta$ is absorbing, $A$ is null recurrent, and $B$ is positive recurrent. (Draw the transition diagram.)