Markov Chains and Return Times

Question

Let $(X_n)_{n≥0}$ be a Markov chain with transition kernel $p$ on a countable state space $S$, starting at $x∈S$

$T^{(1)}=\inf\{n≥1:X_n=x\} \quad \quad$ first return time to $x$

$T^{(2)}=\inf\{n>T^{(1)}:X_n=x\} \quad \quad$ second return time to $x$

Can we prove $$T^{(2)}=\infty \quad \text{a.s.} \Longrightarrow T^{(1)}=\infty \quad \text{a.s.}$$

Answer 1

Hint: If the chain starts at $x$, the strong Markov property at time $T^{(1)}$ indicates that $T^{(2)}=T^{(1)}+R$, where $R$ is independent of $T^{(1)}$ and with the same distribution. Can you conclude from here?