Abstract
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) = np and wondered “[...] what property of the power np 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.
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) = np and wondered “[...] what property of the power np 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.
| Original language | English |
|---|---|
| Pages (from-to) | 296-303 |
| Number of pages | 8 |
| Journal | Electronic Communications in Probability |
| Volume | 16 |
| DOIs | |
| Publication status | Published - 2011 |