Condition for recurrence and transience of MC

[alphabeta89]

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

How do we approach this question? I was thinking of using the theorem below.Suppose $$S$$ is irreducible, and $$\phi\geq 0$$ with $$E_x\phi(X_1) \leq \phi(x)$$ for
$$x\notin F$$, a finite set, and $$\phi(x)\rightarrow \infty$$ as $$x\rightarrow \infty$$, i.e., $${\{x : \phi(x) \leq M}\}$$ is finite for any $$M < \infty$$, then the chain is recurrent.However I have no idea of how to start. Thanks in advance.

 

[mmwave]

To approach this question, we can first define what an irreducible Markov Chain (M.C.) is. An M.C. is said to be irreducible if it is possible to get from any state to any other state in a finite number of steps. In other words, there are no isolated states in the M.C. and every state is accessible from every other state.

In this case, we can show that the given model is indeed irreducible by considering the transition probabilities. Since X_{n+1} is dependent on X_n, we can see that there is a non-zero probability of moving from any state to any other state in one step. This means that the chain is irreducible.

Next, we can consider the recurrence of the chain. Recurrence refers to the tendency of the chain to return to a certain state after a certain number of steps. In this case, we can use the theorem provided in the forum post to show that the chain is recurrent if 0\leq b <1.

To do this, we can define \phi(x) as the expected number of steps it takes to return to state x. From the given model, we can see that the expected number of steps to return to state x is \frac{1}{\lambda}=\frac{1}{a+bX_n}. Since a>0, we can see that \phi(x) \rightarrow \infty as x\rightarrow \infty. This means that the chain is recurrent.

However, if b\geq 1, then \phi(x) does not tend to infinity as x\rightarrow \infty. This means that the chain is not recurrent and is instead transient. In a transient chain, there is a non-zero probability of never returning to a certain state, which is the case when b\geq 1 in this model.

In conclusion, we can approach this question by first defining what an irreducible M.C. is and then using the given model to show that it is indeed irreducible. Next, we can use the provided theorem to show that the chain is recurrent if 0\leq b <1 and transient if b\geq 1.