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Couplings of markov chains by randomized stopping times
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  • Published: 01 June 1987

Couplings of markov chains by randomized stopping times

Part I: Couplings, harmonic functions and the poisson equation

  • Andreas Greven1 

Probability Theory and Related Fields volume 75, pages 195–212 (1987)Cite this article

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Summary

We consider a Markov chain on (E, ℬ) generated by a Markov kernel P. We study the question, when we can find for two initial distributions ν and μ two randomized stopping times T of (νX n ) n∈N and S of ( μ X n ) n∈N , such that the distribution of ν X T equals the one of μ X S and T, S are both finite.

The answer is given in terms of 〈ν-μ, h〉 with h bounded harmonic, or in terms of \(\mathop {\lim }\limits_{n \to \infty } \left\| {\frac{1}{{n + 1}} \cdot \sum\limits_0^n {(v - \mu )P^k } } \right\|\).

For stopping times T, S for two chains ( ν X n ) n∈N ,( μ X n ) n∈N we consider measures η, ζ on (E, ℬ) defined as follows: η(A)=expected number of visits of ( ν X n ) toA before T, ζ(A)=expected number of visits of (μ X n ) toA before S.

We show that we can construct T, S such that η and ζ are mutually singular and ℒ( v X T )=ℒ(μ X S . We relate η and ζ to the positive and negative part of certain solutions of the Poisson equation (I-P)(·)=ν-μ.

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References

  1. Dinges, H.: Stopping sequences. Séminaire de Probabilitées VIII. Lect. Notes Math. 381, 27–36 Berlin Heidelberg New York: Springer 1974

    Google Scholar 

  2. Dinges, H.: The renewal theorem from the point of view of stopping sequences. Proceedings of the Sixth Conference on Probability Theory. Brasov 1979

  3. Derriennic, Y.: Lois ≪zéro ou deux≫ pour les processus de Markov Application aux marches aléatoires. Ann. Inst. H. Poincaré 12, 111–130 (1976)

    MathSciNet  MATH  Google Scholar 

  4. Foguel, R.S.: The ergodic theory of Markov processes. New York: Van Nostrand 1969

    MATH  Google Scholar 

  5. Greven, A.: Stoppfolgen und Kopplungen von Markovprozessen. Dissertation. Frankfurt 1982

  6. Meyer, P.A.: Solutions de l'équation de Poisson dans le cas recurrent. Séminaire de Probabilitées V. Lect. Notes Math. 191, 251–269. Berlin Heidelberg New York: Springer 1971

    Google Scholar 

  7. Neveu, J.: Potentiel markovien récurrent des, chaines de Harris. Ann. Inst. Fourier, Grenoble, 22, 85–130 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  8. Revuz, D.: Remarks on the Filling scheme for recurrent Markov chains. Duke Math. Journal 45, 681–689 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  9. Rost, H.: Markov-Ketten bei sich füllenden Löchern im Zustandsraum. Ann. Inst. Fourier 21, 253–270 (1971)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 294, D-6900, Heidelberg 1, Germany

    Andreas Greven

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  1. Andreas Greven
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Greven, A. Couplings of markov chains by randomized stopping times. Probab. Th. Rel. Fields 75, 195–212 (1987). https://doi.org/10.1007/BF00354033

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  • Received: 23 March 1983

  • Revised: 15 October 1984

  • Published: 01 June 1987

  • Issue Date: June 1987

  • DOI: https://doi.org/10.1007/BF00354033

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Keywords

  • Markov Chain
  • Stochastic Process
  • Probability Theory
  • Statistical Theory
  • Poisson Equation
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