Consider a continuous-time Markov chain $\left(X_{t}\right)$ given by the diagram below.

We will assume that the rates $\alpha, \beta, \lambda$ and $\mu$ are all positive.

(a) Is the chain $\left(X_{t}\right)$ irreducible?

(b) Write down the standard equations for the hitting probabilities

$$h_{\mathrm{C} i}=\mathbb{P}_{\mathrm{C} i}(\text { hit W0) }, \quad i \geqslant 0,$$

and

$$h_{\mathrm{W} i}=\mathbb{P}_{\mathrm{W} i}(\text { hit W0) }, \quad i \geqslant 1 .$$

Explain how to identify the probabilities $h_{\mathrm{C} i}$ and $h_{\mathrm{W} i}$ among the solutions to these equations.

[You should state the theorem you use but its proof is not required.]

(c) Set $h^{(i)}=\left(\begin{array}{c}h_{\mathrm{C} i} \\ h_{\mathrm{W} i}\end{array}\right)$ and find a matrix $A$ such that

$$h^{(i)}=A h^{(i-1)}, \quad i=1,2, \ldots$$

The recursion matrix $A$ has a 'standard' eigenvalue and a 'standard' eigenvector that do not depend on the transition rates: what are they and why are they always present?

(d) Calculate the second eigenvalue $\vartheta$ of the matrix $A$, and the corresponding eigenvector, in the form $\left(\begin{array}{l}b \\ 1\end{array}\right)$, where $b>0$.

(e) Suppose the second eigenvalue $\vartheta$ is $\geqslant 1$. What can you say about $h_{\mathrm{C} i}$ and $h_{\mathrm{W} i}$ ? Is the chain $\left(X_{t}\right)$ transient or recurrent? Justify your answer.

(f) Now assume the opposite: the second eigenvalue $\vartheta$ is $<1$. Check that in this case $b<1$. Is the chain transient or recurrent under this condition?

(g) Finally, specify, by means of inequalities between the parameters $\alpha, \beta, \lambda$ and $\mu$, when the chain $\left(X_{t}\right)$ is recurrent and when it is transient.