Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-25T17:25:25.300Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic domination and Markovian couplings

Part of: Markov processes Special processes

Published online by Cambridge University Press:  01 July 2016

F. Javier López ,
Servet Martínez  and
Gerardo Sanz
F. Javier López*
Affiliation:
Universidad de Zaragoza
Servet Martínez*
Affiliation:
Universidad de Chile
Gerardo Sanz*
Affiliation:
Universidad de Zaragoza
*
∗ Postal address: Departamento de Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, C/Pedro Cerbuna 12, 50009 Zaragoza, Spain.
∗∗ Postal address: Departamento de Ingeniería Matemática y Centro de Modelamiento Matemático UMR 2071, CNRS, Universidad de Chile, Casilla 170-3, Correo 3, Santiago, Chile. Email address: [email protected]
∗ Postal address: Departamento de Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, C/Pedro Cerbuna 12, 50009 Zaragoza, Spain.
Get access Rights & Permissions [Opens in a new window]

Abstract

For continuous-time Markov chains with semigroups P, P' taking values in a partially ordered set, such that P ≤ stP', we show the existence of an order-preserving Markovian coupling and give a way to construct it. From our proof, we also obtain the conditions of Brandt and Last for stochastic domination in terms of the associated intensity matrices. Our result is applied to get necessary and sufficient conditions for the existence of Markovian couplings between two Jackson networks.

Keywords

Markov chainorder-preserving couplingHahn-Banach theoremJackson network

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60K25: Queueing theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 32 , Issue 4 , December 2000 , pp. 1064 - 1076
Copyright
Copyright © Applied Probability Trust 2000 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Anderson, W. J. (1991). Continuous-time Markov chains. An applications-oriented approach. Springer, New York.Google Scholar
[2] Brandt, A. and Last, G. (1994). On the pathwise comparison of jump processes driven by stochastic intensities. Math. Nachr. 167, 21–42.CrossRefGoogle Scholar
[3] Chen, M. F. (1992). From Markov Chains to Non-equilibrium Particle Systems. World Scientific, Singapore.Google Scholar
[4] Kamae, T., Krengel, G. and O'Brien, G. L. (1977). Stochastic inequalities on partially ordered spaces. Ann. Prob. 5, 899–912.Google Scholar
[5] Keilson, J. and Kester, A. (1977). Monotone matrices and monotone Markov processes. Stoch. Proc. Appl. 5, 231–241.Google Scholar
[6] Liggett, T. M. (1985). Interacting Particle Systems. Springer, New York.Google Scholar
[7] Lindvall, T. (1992). Lectures on the Coupling Method. John Wiley, New York.Google Scholar
[8] Lindvall, T. (1997). Stochastic monotonicities in Jackson queueing. Prob. Eng. Inf. Sci. 11, 1–9.Google Scholar
[9] Lindvall, T. (1999). On Strassen's theorem on stochastic domination. Electron. Comm. Prob. 4, 51–59.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] Massey, W.A. (1987). Stochastic orderings for Markov processes on partially ordered spaces. Math. Operat. Res. 40, 350–367.Google Scholar
[12] Rockafellar, R.T. (1972). Convex Analysis, 2nd edn. Princeton University Press.Google Scholar
[13] Rudin, W. (1973). Functional Analysis. McGraw-Hill, New York.Google Scholar
[14] Stoyan, D. (1983). Comparison Methods for Queues and Other Stochastic Models. John Wiley, New York.Google Scholar
[15] Strassen, V. (1965). The existence of probability measures with given marginals. Ann. Math. Statist. 36, 423–439.CrossRefGoogle Scholar
[16] Thorisson, H. (2000). Coupling, Stationarity and Regeneration. Springer, New York.CrossRefGoogle Scholar