Electronic Communications in Probability

Moments of recurrence times for Markov chains

Frank Aurzada, Hanna Döring, Marcel Ortgiese, and Michael Scheutzow

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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.

Article information

Source
Electron. Commun. Probab., Volume 16 (2011), paper no. 28, 296-303.

Dates
Accepted: 8 June 2011
First available in Project Euclid: 7 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465261984

Digital Object Identifier
doi:10.1214/ECP.v16-1632

Mathematical Reviews number (MathSciNet)
MR2811181

Zentralblatt MATH identifier
1231.60063

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords
Discrete time Markov chain recurrence time generalized moment

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Aurzada, Frank; Döring, Hanna; Ortgiese, Marcel; Scheutzow, Michael. Moments of recurrence times for Markov chains. Electron. Commun. Probab. 16 (2011), paper no. 28, 296--303. doi:10.1214/ECP.v16-1632. https://projecteuclid.org/euclid.ecp/1465261984


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References

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