Let $\left(X_{n}\right)_{n \geqslant 0}$ be an irreducible Markov chain with $p_{i j}^{(n)}=P\left(X_{n}=j \mid X_{0}=i\right)$. Define the meaning of the statements:

(i) state $i$ is transient,

(ii) state $i$ is aperiodic.

Give a criterion for transience that can be expressed in terms of the probabilities $\left(p_{i i}^{(n)}, n=0,1, \ldots\right)$.

Prove that if a state $i$ is transient then all states are transient.

Prove that if a state $i$ is aperiodic then all states are aperiodic.

Suppose that $p_{i i}^{(n)}=0$ unless $n$ is divisible by 3 . Given any other state $j$, prove that $p_{j j}^{(n)}=0$ unless $n$ is divisible by 3 .