Coupling of Markov chains by randomized stopping times. I: Couplings, harmonic functions and the Poisson equation
type: Item
instance of: scholarly article
title: Coupling of Markov chains by randomized stopping times. I: Couplings, harmonic functions and the Poisson equation
zbMATH Open document ID: 0596.60070
DOI: 10.1007/BF00354033
publication date: 1987-01-01T00:00:00Z
review text: We consider a Markov chain on (E,\({\mathcal B})\) generated by a Markov kernel P. We study the question, when we can find for two initial distributions \(\nu\) and \(\mu\) two randomized stopping times T of \((_{\nu}X_ n)_{n\in N}\) and S of \((_{\mu}X_ n)_{n\in N}\), such that the distribution of \(_{\nu}X_ T\) equals the one of \(_{\mu}X_ S\) and T, S are both finite. The answer is given in terms of \(<\nu -\mu,h>\) with h bounded harmonic, or in terms of \(\lim_{n\to \infty}\| 1/(n+1)\sum^{n}_{0}(\nu - \mu)P^ k\|\). For stopping times S, T for two chains \((_{\nu}X_ n)_{n\in N}\), \((_{\mu}X_ n)_{n\in N}\) we consider measures \(\eta\), \(\xi\) on (E,\({\mathcal B})\) defined as follows: \(\eta (A)= \exp ected\) number of visits of \((_{\nu}X_ n)\) to A before T, \(\xi (A)= \exp ected\) number of visits of \((_{\mu}X_ n)\) to A before S. We show that we can construct S, T such that \(\eta\) and \(\xi\) are mutually singular and \({\mathcal L}(_{\nu}X_ T)={\mathcal L}(_{\mu}X_ S)\). We relate \(\eta\) and \(\xi\) to the positive and negative part of certain solutions of the Poisson equation \((I-P)(\cdot)=\nu -\mu.\)
Mathematics Subject Classification ID: 60J10
Mathematics Subject Classification ID: 60G30
zbMATH DE Number: 3960702
zbMATH Keywords: randomized stopping times
zbMATH Keywords: Poisson equation
zbMATH Keywords: coupling of Markov chains
author: Andreas Greven
MaRDI profile type: Publication
cites work: Q4775243
cites work: Q3920400
cites work: Q4124070
cites work: Q4768403
cites work: Q3316339
cites work: Potentiel markovien récurrent des chaînes de Harris. (Recurrent Markov potential of Harris chains)
cites work: Remarks on the filling scheme for recurrent Markov chains
cites work: Markov-Ketten bei sich füllenden Löchern im Zustandsraum
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
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description: scientific article; zbMATH DE number 3960702