We consider moments of the return times (or first hitting times) in an irreducible discrete time discrete space Markov chain. It is classical that the finiteness of the first moment of a return time of one state implies the finiteness of the first moment of the first return time of any other state. We extend this statement to moments with respect to a function $f$, where $f$ satisfies a certain, best possible condition. This generalizes results of K.L. Chung (1954) who considered the functions $f(n)=n^p$ and wondered "[...] what property of the power $n^p$ lies behind this theorem [...]" (see Chung (1967), p. 70). We exhibit that exactly the functions that do not increase exponentially - neither globally nor locally - fulfill the above statement.
"Moments of recurrence times for Markov chains." Electron. Commun. Probab. 16 296 - 303, 2011. https://doi.org/10.1214/ECP.v16-1632