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Markovian couplings staying in arbitrary subsets of the state space

Part of: Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

F. Javier López  and
Gerardo Sanz
F. Javier López*
Affiliation:
Universidad de Zaragoza
Gerardo Sanz*
Affiliation:
Universidad de Zaragoza
*
∗ Postal address: Dpto. Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain.
∗ Postal address: Dpto. Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain.
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Abstract

Let (Xt) and (Yt) be continuous-time Markov chains with countable state spaces E and F and let K be an arbitrary subset of E x F. We give necessary and sufficient conditions on the transition rates of (Xt) and (Yt) for the existence of a coupling which stays in K. We also show that when such a coupling exists, it can be chosen to be Markovian and give a way to construct it. In the case E=F and K ⊆ E x E, we see how the problem of construction of the coupling can be simplified. We give some examples of use and application of our results, including a new concept of lumpability in Markov chains.

Keywords

Markov chaincouplinglumpability

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60K25: Queueing theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 39 , Issue 1 , March 2002 , pp. 197 - 212
Copyright
Copyright © Applied Probability Trust 2002 

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References

[1]. Ahuja, R. K., Magnanti, T. L., and Orlin, J. B. (1993). Network Flows. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
[2]. Ball, F., and Yeo, G. F. (1993). Lumpability and marginalisability for continuous-time Markov chains. J. Appl. Prob. 30, 518–528.CrossRefGoogle Scholar
[3]. Buchholz, P. (1994). Exact and ordinary lumpability in finite Markov chains. J. Appl. Prob. 31, 59–75.Google Scholar
[4]. Dempster, A. P. (1967). Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Statist. 38, 325–339.Google Scholar
[5]. Doisy, M. (2000). A coupling technique for stochastic comparison of functions of Markov processes. J. Appl. Math. Decis. Sci. 4, 39–64.CrossRefGoogle Scholar
[6]. Kamae, T., Krengel, G., and O’Brien, G. L. (1977). Stochastic inequalities on partially ordered spaces. Ann. Prob. 5, 899–912.Google Scholar
[7]. Kemeny, J. G., and Snell, J. L. (1976). Finite Markov Chains. Springer, New York.Google Scholar
[8]. Lindvall, T. (1992). Lectures on the Coupling Method. John Wiley, New York.Google Scholar
[9]. Lindvall, T. (1997). Stochastic monotonicities in Jackson queueing networks. Prob. Eng. Inf. Sci. 11, 1–9.Google Scholar
[10]. López, F. J., and Sanz, G. (1998). Stochastic comparisons and couplings for interacting particle systems. Statist. Prob. Lett. 40, 93–102.Google Scholar
[11]. López, F. J., and Sanz, G. (2000). Stochastic comparisons for general probabilistic cellular automata. Statist. Prob. Lett. 46, 401–410.CrossRefGoogle Scholar
[12]. López, F. J., Martínez, S., and Sanz, G. (2000). Stochastic domination and Markovian couplings. Adv. Appl. Prob. 32, 1064–1076.CrossRefGoogle Scholar
[13]. Martínez, S., and Ycart, B. (2001). Decay rates and cutoff for convergence and hitting times of Markov chains with countably infinite state space. Adv. Appl. Prob. 33, 188–205.CrossRefGoogle Scholar
[14]. Massey, W. A. (1987). Stochastic orderings for Markov processes on partially ordered spaces. Math. Operat. Res. 12, 350–367.Google Scholar
[15]. Rubino, G., and Sericola, B. (1989). On weak lumpability in Markov chains. J. Appl. Prob. 26, 446–457.CrossRefGoogle Scholar
[16]. Stoyan, D. (1983). Comparison Methods for Queues and Other Stochastic Models. John Wiley, New York.Google Scholar
[17]. Thorrison, H. (2000). Coupling, Stationarity and Regeneration. Springer, New York.Google Scholar
[18]. Whitt, W. (1986). Stochastic comparisons for non-Markov processes. Math. Operat. Res. 11, 608–618.CrossRefGoogle Scholar