Consider an irreducible ergodic Markov chain on a finite state space $S$. Then any state is positive recurrent and this should suffice to conclude that the mean hitting time of state $s \in S$ is finite. Is this reasoning correct?
What can be said, in general, about the second (or higher) moment of such hitting time? Is it always finite? If not, which additional conditions guarantee that it is finite?
Are the answers still the same if we consider an irreducible ergodic Markov process (i.e. in continuous time), always on a finite state space?
To show this in the discrete time case, first note that, for every state $x$, the hitting time $T$ of $s$ starting from $x$ is almost surely finite because $x$ and $s$ communicate and the state space is finite. Hence there exists some finite $n_x$ such that $P[T\gt n_x\mid X_0=x]\leqslant\mathrm e^{-1}$. Choose any finite $n$ such that $P[T\gt n\mid X_0=x]\leqslant\mathrm e^{-1}$ for every $x$, for example $n=\max\{n_x\mid x\ \text{state}\}$. Note that this maximum exists (and is finite), once again because the state space is finite. For every $k\geqslant0$, $$ P[T\gt k+n]=\sum_xP[T\gt k+n,X_k=x,T\gt k], $$ hence $$ P[T\gt k+n]=\sum_xP[T\gt k+n\mid X_k=x,T\gt k]\cdot P[X_k=x,T\gt k]. $$ For every $x$, $P[T\gt k+n\mid X_k=x,T\gt k]=P[T\gt n\mid X_0=x]$ by the (simple) Markov property at time $k$, hence $$ P[T\gt k+n]\leqslant\sum_x\mathrm e^{-1}\cdot P[X_k=x,T\gt k]=\mathrm e^{-1}\cdot P[T\gt k]. $$ Iterating this, one gets $P[T\gt in]\leqslant\mathrm e^{-i}$ for every $i$, hence $P[T\gt k]\leqslant\mathrm e^{1-(k/n)}$ for every $k$. In particular, $E[\mathrm e^{aT}]$ is finite for every positive $a\lt1/n$.
The continuous time finite state space case is similar.