Ergodicity and Positive Recurrence

Question

If a stochastic process $\{ X_t \}_{t \in \mathbb{N}}$, that takes values only in $\mathcal{X} = \{0, 1\}$, is ergodic, then what can we say about its recurrence times? If $T_k$ is the $k$th recurrence time for the state $1$, then is $\mathbb{E}[T_k]$ finite? What about $\mathbb{E}[T^{2}_{k}]$, or other moments of $T_{k}$?