A Markov chain $\left(X_{n}\right)_{n \geqslant 0}$ has as its state space the integers, with

$$p_{i, i+1}=p, \quad p_{i, i-1}=q=1-p$$

and $p_{i j}=0$ otherwise. Assume $p>q$.

Let $T_{j}=\inf \left\{n \geqslant 1: X_{n}=j\right\}$ if this is finite, and $T_{j}=\infty$ otherwise. Let $V_{0}$ be the total number of hits on 0 , and let $V_{0}(n)$ be the total number of hits on 0 within times $0, \ldots, n-1$. Let

$$\begin{aligned} h_{i} &=P\left(V_{0}>0 \mid X_{0}=i\right) \\ r_{i}(n) &=E\left[V_{0}(n) \mid X_{0}=i\right] \\ r_{i} &=E\left[V_{0} \mid X_{0}=i\right] \end{aligned}$$

(i) Quoting an appropriate theorem, find, for every $i$, the value of $h_{i}$.

(ii) Show that if $\left(x_{i}, i \in \mathbb{Z}\right)$ is any non-negative solution to the system of equations

$$\begin{aligned} x_{0} &=1+q x_{1}+p x_{-1}, \\ x_{i} &=q x_{i-1}+p x_{i+1}, \quad \text { for all } i \neq 0 \end{aligned}$$

then $x_{i} \geqslant r_{i}(n)$ for all $i$ and $n$.

(iii) Show that $P\left(V_{0}\left(T_{1}\right) \geqslant k \mid X_{0}=1\right)=q^{k}$ and $E\left[V_{0}\left(T_{1}\right) \mid X_{0}=1\right]=q / p$.

(iv) Explain why $r_{i+1}=(q / p) r_{i}$ for $i>0$.

(v) Find $r_{i}$ for all $i$.