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Continuity properties of a factor of Markov chains

Published online by Cambridge University Press:  24 March 2016

Walter A. F. de Carvalho ,
Sandro Gallo  and
Nancy L. Garcia
Walter A. F. de Carvalho
Affiliation:
Alameda Pitangueiras, 231, Engenheiro Coelho, São Paulo, 13.165-000, Brazil. Email address: [email protected]
Sandro Gallo
Affiliation:
Departamento de Estatística - UFSCar, Rodovia Washington Luís, s/n - Jardim Guanabara, São Carlos, São Paulo, 13565-905, Brazil. Email address: [email protected]
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Abstract

Starting from a Markov chain with a finite or a countable infinite alphabet, we consider the chain obtained when all but one symbol are indistinguishable for the practitioner. We study conditions on the transition matrix of the Markov chain ensuring that the image chain has continuous or discontinuous transition probabilities with respect to the past.

Keywords

Factor mapcontinuity ratechains of infinite order
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 1 , March 2016 , pp. 216 - 230
Copyright
Copyright © Applied Probability Trust 2016 

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References

Boyle, M. and Petersen, K. (2011). Hidden Markov processes in the context of symbolic dynamics. In Entropy of Hidden Markov Processes and Connections to Dynamical Systems (London Math. Soc. Lecture Note Ser. 385), Cambridge University Press, pp. 5–71. Google Scholar
Burke, C. J. and Rosenblatt, M. (1958). A Markovian function of a Markov chain. Ann. Math. Statist. 29, 1112–1122. CrossRefGoogle Scholar
Chazottes, J.-R. and Ugalde, E. (2003). Projection of Markov measures may be Gibbsian. J. Statist. Phys. 111, 1245–1272. Google Scholar
Chazottes, J.-R. and Ugalde, E. (2011). On the preservation of Gibbsianness under symbol amalgamation. In Entropy of Hidden Markov Processes and Connections to Dynamical Systems (London Math. Soc. Lecture Note Ser. 385), Cambridge University Press, pp. 72–97. Google Scholar
Darroch, J. N. and Seneta, E. (1967). On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J. Appl. Prob. 4, 192–196. Google Scholar
Fernández, R. and Maillard, G. (2004a). Chains and specifications. Markov Process. Relat. Fields 10, 435–456. Google Scholar
Fernández, R. and Maillard, G. (2004b). Chains with complete connections and one-dimensional Gibbs measures. Electron. J. Prob. 9, 145–176. CrossRefGoogle Scholar
Fernández, R., Gallo, S. and Maillard, G. (2011). Regular C-measures are not always Gibbsian. Electron. Commun. Prob. 16, 732–740. Google Scholar
Foss, S. G. and Tweedie, R. L. (1998). Perfect simulation and backward coupling. Commun. Statist. Stoch. Models 14, 187–203. CrossRefGoogle Scholar
Harris, T. E. (1955). On chains of infinite order. Pacific J. Math. 5, 707–724. CrossRefGoogle Scholar
Kalikow, S. (1990). Random Markov processes and uniform martingales. Israel J. Math. 71, 33–54. Google Scholar
Kitchens, B. P. (1998). Symbolic Dynamics. Springer, Berlin. Google Scholar
Pollicott, M. and Kempton, T. (2011). Factors of Gibbs measures for full shifts. In Entropy of Hidden Markov Processes and Connections to Dynamical Systems (London Math. Soc. Lecture Note Ser. 385), Cambridge University Press, pp. 246–257. CrossRefGoogle Scholar
Redig, F. and Wang, F. (2010). Transformations of one-dimensional Gibbs measures with infinite range interaction. Markov Process. Relat. Fields 16, 737–752. Google Scholar
Seneta, E. (2006). Non-Negative Matrices and Markov Chains, 2nd edn. Springer, New York. Google Scholar
Verbitskiy, E. (2011a). On factors of g-measures. Indag. Math. (N.S.) 22, 315–329. Google Scholar
Verbitskiy, E. (2011b). Thermodynamics of hidden Markov processes. In Entropy of Hidden Markov Processes and Connections to Dynamical Systems (London Math. Soc. Lecture Note Ser. 385), Cambridge University Press, pp. 258–272. Google Scholar
Yoo, J. (2010). On factor maps that send Markov measures to Gibbs measures. J. Statist. Phys. 141, 1055–1070. CrossRefGoogle Scholar