A Note on Independence Copula for Conditional Markov Chains

  • Tomasz R. BieleckiEmail author
  • Jacek Jakubowski
  • Mariusz Niewęgłowski
Part of the Fields Institute Communications book series (FIC, volume 79)


Given a family (Y k ,  k = 1, 2, , N) of conditional Markov chains, we construct a conditional Markov chain X = (X 1, , X N ) such that X k , k = 1, 2, , N, are conditional Markov chains, which are conditionally independent given the information contained in some filtration \(\mathbb{F}\), and such that for each k the conditional law of X k coincides with the conditional law of Y k . This is a new result that can be used to model different phenomena such as the gating behavior of multiple ion channels in a membrane patch, or credit ratings migrations.



We thank the referees and the editors for valuable comments and suggestions, which we used revising the original version of this note.

Research of T.R. Bielecki was partially supported by NSF grant DMS-1211256.


  1. 1.
    Ball, F., Milne, R.K., Yeo, G.F.: Continuous-time Markov chains in a random environment, with applications to ion channel modelling. Adv. in Appl. Probab. 26(4), 919–946 (1994). DOI 10.2307/1427898. URL MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ball, F., Yeo, G.F.: Lumpability and marginalisability for continuous-time Markov chains. J. Appl. Probab. 30(3), 518–528 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Biagini, F., Groll, A., Widenmann, J.: Intensity-based premium evaluation for unemployment insurance products. Insurance Math. Econom. 53(1), 302–316 (2013). DOI 10.1016/j.insmatheco.2013.06.001. URL MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bielecki, T.R., Jakubowski, J., Niewęgłowski, M.: Intricacies of dependence between components of multivariate Markov chains: weak Markov consistency and weak Markov copulae. Electron. J. Probab. 18, no. 45, 21 (2013). DOI 10.1214/EJP.v18-2238. URL
  5. 5.
    Bielecki, T.R., Jakubowski, J., Niewęgłowski, M.: Conditional Markov chains, part I: construction and properties (2015). URL
  6. 6.
    Bielecki, T.R., Jakubowski, J., Niewęgłowski, M.: Conditional Markov chains, part II: consistency and copulae (2015). URL
  7. 7.
    Dabrowski, A.R., McDonald, D.: Statistical analysis of multiple ion channel data. Ann. Statist. 20(3), 1180–1202 (1992). DOI 10.1214/aos/1176348765. URL MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Horn, R.A., Johnson, C.R.: Topics in matrix analysis. Cambridge University Press, Cambridge (1994). Corrected reprint of the 1991 originalGoogle Scholar
  9. 9.
    Jakubowski, J., Niewęgłowski, M.: A class of \(\mathbb{F}\)-doubly stochastic Markov chains. Electron. J. Probab. 15, no. 56, 1743–1771 (2010). DOI 10.1214/EJP.v15-815. URL MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jakubowski, J., Pytel, A.: The Markov consistency of Archimedean survival processes. J. Appl. Probab. 53(2), 293–409 (2015)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kijima, S., Kijima, H.: Statistical analysis of channel current from a membrane patch. I. Some stochastic properties of ion channels or molecular systems in equilibrium. J. Theoret. Biol. 128(4), 423–434 (1987). DOI 10.1016/S0022-5193(87)80188-1. URL zbMATHGoogle Scholar

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© Springer Science+Business Media LLC 2017

Authors and Affiliations

  • Tomasz R. Bielecki
    • 1
    Email author
  • Jacek Jakubowski
    • 2
    • 3
  • Mariusz Niewęgłowski
    • 3
  1. 1.Department of Applied MathematicsIllinois Institute of TechnologyChicagoUSA
  2. 2.Institute of MathematicsUniversity of WarsawWarszawaPoland
  3. 3.Faculty of Mathematics and Information ScienceWarsaw University of TechnologyWarszawaPoland

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