Let $\left(X_{n}: n \geqslant 0\right)$ be a homogeneous Markov chain with state space $S$ and transition $\operatorname{matrix} P=\left(p_{i, j}: i, j \in S\right)$.

(a) Let $W_{n}=X_{2 n}, n=0,1,2, \ldots$ Show that $\left(W_{n}: n \geqslant 0\right)$ is a Markov chain and give its transition matrix. If $\lambda_{i}=\mathbb{P}\left(X_{0}=i\right), i \in S$, find $\mathbb{P}\left(W_{1}=0\right)$ in terms of the $\lambda_{i}$ and the $p_{i, j}$.

[Results from the course may be quoted without proof, provided they are clearly stated.]

(b) Suppose that $S=\{-1,0,1\}, p_{0,1}=p_{-1,-1}=0$ and $p_{-1,0} \neq p_{1,0}$. Let $Y_{n}=\left|X_{n}\right|$, $n=0,1,2, \ldots$ In terms of the $p_{i, j}$, find

(i) $\mathbb{P}\left(Y_{n+1}=0 \mid Y_{n}=1, Y_{n-1}=0\right)$ and

(ii) $\mathbb{P}\left(Y_{n+1}=0 \mid Y_{n}=1, Y_{n-1}=1, Y_{n-2}=0\right)$.

What can you conclude about whether or not $\left(Y_{n}: n \geqslant 0\right)$ is a Markov chain?