(i) Let $\left(X_{n}, Y_{n}\right)_{n \geqslant 0}$ be a simple symmetric random walk in $\mathbb{Z}^{2}$, starting from $(0,0)$, and set $T=\inf \left\{n \geqslant 0: \max \left\{\left|X_{n}\right|,\left|Y_{n}\right|\right\}=2\right\}$. Determine the quantities $\mathbb{E}(T)$ and $\mathbb{P}\left(X_{T}=2\right.$ and $\left.Y_{T}=0\right)$.

(ii) Let $\left(X_{n}\right)_{n \geqslant 0}$ be a discrete-time Markov chain with state-space $I$ and transition matrix $P$. What does it mean to say that a state $i \in I$ is recurrent? Prove that $i$ is recurrent if and only if $\sum_{n=0}^{\infty} p_{i i}^{(n)}=\infty$, where $p_{i i}^{(n)}$ denotes the $(i, i)$ entry in $P^{n}$.

Show that the simple symmetric random walk in $\mathbb{Z}^{2}$ is recurrent.