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Condition for recurrence and transience of MC

alphabeta89

New member
Apr 16, 2013
2
Consider the following model.


[TEX] X_{n+1}[/TEX] given [TEX]X_n, X_{n-1},...,X_0[/TEX] has a Poisson distribution with mean [TEX]\lambda=a+bX_n[/TEX] where [TEX]a>0,b\geq{0}[/TEX]. Show that [TEX]X=(X_n)_{n\in\mathrm{N_0}}[/TEX] is an irreducible M.C & it is recurrent if [TEX]0\leq b <1[/TEX]. In addition, it is transient if [TEX]b\geq 1[/TEX].

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


Suppose [TEX]S[/TEX] is irreducible, and [TEX]\phi\geq 0[/TEX] with [TEX]E_x\phi(X_1) \leq \phi(x)[/TEX] for
[TEX]x\notin F[/TEX], a finite set, and [TEX]\phi(x)\rightarrow \infty[/TEX] as [TEX]x\rightarrow \infty[/TEX], i.e., [TEX]{\{x : \phi(x) \leq M}\}[/TEX] is finite for any [TEX]M < \infty[/TEX], then the chain is recurrent.


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