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Moments of recurrence times for Markov chains

Authors
  • Aurzada, Frank
  • Doering, Hanna
  • Ortgiese, Marcel
  • Scheutzow, Michael
Type
Published Article
Publication Date
Apr 11, 2011
Submission Date
Apr 11, 2011
Source
arXiv
License
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External links

Abstract

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.

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