Consider a branching process with offspring distribution given by $\{p_n\}$. We will make the process into an irreducible Markov chain by asserting that if the population ever dies out, then the next generation will have one new individual [in other words, $p(0,1) = 1$]. For which $\{p_n\}$ will this chain be positive recurrent, null recurrent, transient?
This is an exercise from Lawler's Introduction to Stochastic Processes.
I believe the result should be (let $\mu=\sum_{i=0}^{\infty}ip_i$): If $\mu < 1$, it is positive recurrent; if $\mu = 1$, it is null recurrent; if $\mu > 1$, it is transient. But I don't know how to prove it. Thanks for help!