Let $\left(X_{n}\right)_{n \geqslant 0}$ be a Markov chain on a state space $S$, and let $p_{i j}(n)=\mathbb{P}\left(X_{n}=j \mid X_{0}=i\right)$.

(i) What does the term communicating class mean in terms of this chain?

(ii) Show that $p_{i i}(m+n) \geqslant p_{i j}(m) p_{j i}(n)$.

(iii) The period $d_{i}$ of a state $i$ is defined to be

$$d_{i}=\operatorname{gcd}\left\{n \geqslant 1: p_{i i}(n)>0\right\}$$

Show that if $i$ and $j$ are in the same communicating class and $p_{j j}(r)>0$, then $d_{i}$ divides $r$.