Consider the following model.

given has a Poisson distribution with mean where . Show that is an irreducible M.C & it is recurrent if . In addition, it is transient if .

How do we approach this question? I was thinking of using the theorem below.

Suppose is irreducible, and with for

, a finite set, and as , i.e., is finite for any , then the chain is recurrent.

However I have no idea of how to start. Thanks in advance.