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 $f(n)=n^p$ 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.
Highlights
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
A classical result, see e.g. [Kol36], states that for any recurrent, irreducible Markov chain on a countable state space the following holds: if for any state i the first moment of the recurrence time is finite this applies to any other state
If we denote by Ti j the first time that the Markov chain visits state j if it is started in i, the result can be stated as follows: E f (Tii) < ∞ for some state i implies that E f (Tj j) < ∞ for any other state j, where f (x) = x n for some integer n
Summary
A classical result, see e.g. [Kol36], states that for any recurrent, irreducible Markov chain on a countable state space the following holds: if for any state i the first moment of the recurrence time is finite this applies to any other state. 2], factorial moments [Lam60], and more recently, explicit formulas for higher polynomial moments [Sze08] These considerations naturally lead to the question, for which functions it is true that a finite generalized moment of the return time for one state i implies that the moment is finite for the return time for any other state of the Markov chain. We only consider irreducible, recurrent discrete time Markov chains with a countable state space. Our objective is to classify the collection of all f ∈ such that for each irreducible recurrent discrete time Markov chain with a finite or countably infinite state space E, the following holds: if E f (Tii) < ∞ for some i ∈ E E f (Tj j) < ∞ for all j ∈ E. Together with the earlier observation that ⊆ , so that (a) ⇒ (b), this shows the equivalence of the three statements
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