  We consider moments of the return times (or first hitting times) in a
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.
