Electronic Communications in Probability

Moments of recurrence times for Markov chains

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

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Discrete time Markov chain recurrence time generalized moment

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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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