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Dependence ordering for Markov processes on partially ordered spaces

Part of: Special processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Hans Daduna  and
Ryszard Szekli
Hans Daduna*
Affiliation:
Hamburg University
Ryszard Szekli*
Affiliation:
Wrocław University
*
∗Postal address: Department of Mathematics, Hamburg University, Bundesstrasse 55, 20146 Hamburg, Germany.
∗∗Postal address: Mathematical Institute, Wrocław University, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland. Email address: szekli@math.uni.wroc.pl
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Abstract

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We compare dependence in stochastically monotone Markov processes with partially ordered Polish state spaces using the concordance and supermodular orders. We show necessary and sufficient conditions for the concordance order to hold both in terms of the one-step transition probabilities for discrete-time processes and in terms of the corresponding infinitesimal generators for continuous-time processes. We give examples showing that a stochastic monotonicity assumption is not necessary for such orderings. We indicate relations between dependence orderings and, variously, the asymptotic variance-reduction effect in Monte Carlo Markov chains, Cheeger constants, and positive dependence for Markov processes.

Keywords

Concordance ordersupermodular orderstochastic monotonicitymonotone Markov processdependence orderCheeger constantassociationMCMC

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces 60K25: Queueing theory
Type
Research Article
Information
Journal of Applied Probability , Volume 43 , Issue 3 , September 2006 , pp. 793 - 814
Copyright
© Applied Probability Trust 2006 

Footnotes

Supported by the KBN, grant number 2P03A02023.

References

Alon, N. and Millman, V. D. (1985). “λ_1, isoperimetric inequalities for graphs, and superconcentrators.” J. Combinatorial Theory B 38, 73–88.CrossRefGoogle Scholar
Chen, M.-F. (2004).” From Markov Chains to Non-Equilibrium Particle Systems. World Scientific, Singapore.CrossRefGoogle Scholar
Chen, M.-F. (2005).” Eigenvalues, Inequalities and Ergodic Theory. Springer, London.Google Scholar
Chen, M.-F. and Wang, F.-Y. (1993). “On order preservation and positive correlations for multidimensional diffusion processes.” Prob. Theory Relat. Fields 95, 421–428.CrossRefGoogle Scholar
Chen, M.-F. and Wang, F.-Y. (2000). “Cheeger's inequalities for general symmetric forms and existence criteria for spectral gap.” Ann. Prob. 28, 235–257.CrossRefGoogle Scholar
Christofides, T. C. and Vaggelatou, E. (2004). “A connection between supermodular ordering and positive/negative association.” J. Multivariate Anal. 88, 138–151.CrossRefGoogle Scholar
Daduna, H. and Szekli, R. (1995). “Dependencies in Markovian networks.” Adv. Appl. Prob. 27, 226–254.CrossRefGoogle Scholar
Daley, D. J. (1968). “The correlation structure of the output process of some single server queueing systems.” Ann. Math. Statist. 39, 1007–1019.CrossRefGoogle Scholar
Diaconis, P. and Stroock, D. (1991). “Geometric bounds for eigenvalues of Markov chains.” Ann. Appl. Prob. 1, 36–61.CrossRefGoogle Scholar
Griffeath, D. (1979).” Additive and Cancellative Interacting Particle Systems (Lecture Notes Math. 724). Springer, Berlin.CrossRefGoogle Scholar
Hardy, G. H., Littlewood, J. E. and Polya, G. (1952).” Inequalities, 2nd edn. ” Cambridge University Press.Google Scholar
Hoeffding, W. (1940). “Massstabinvariante Korrelationstheorie.” Schr. Math. Inst. Inst. Angew. Math. Univ. Berlin 5, 179–233.Google Scholar
Hu, T. and Pan, X. (2000). “Comparisons of dependence for stationary Markov processes.” Prob. Eng. Inf. Sci. 14, 299–315.CrossRefGoogle Scholar
Hu, T., Müller, A. and Scarsini, M. (2004). “Some counterexamples in positive dependence.” J. Statist. Planning Infer. 124, 153–158.CrossRefGoogle Scholar
Joe, H. (1990). “Multivariate concordance.” J. Multivariate Anal. 35, 12–30.CrossRefGoogle Scholar
Joe, H. (1997).” Multivariate Models and Dependence Concepts. Chapman and Hall, London.Google Scholar
Keilson, J. and Kester, A. (1977). “Monotone matrices and monotone Markov processes.” Stoch. Process. Appl. 5, 231–241.CrossRefGoogle Scholar
Kulik, R. and Szekli, R. (2004). “Dependence orderings for some functionals of multivariate point processes.” J. Multivariate Anal. 92, 145–173.CrossRefGoogle Scholar
Li, H. and Xu, S. H. (2000). “Stochastic bounds and dependence properties of survival times in a multicomponent shock model.” J. Appl. Prob. 37, 1020–1043.CrossRefGoogle Scholar
Liggett, T. M. (1985).” Interacting Particle Systems. Springer, New York.CrossRefGoogle Scholar
Lindqvist, B. H. (1988). “Association of probability measures.” J. Multivariate Anal. 26, 111–132.CrossRefGoogle Scholar
Lorentz, G. G. (1953). “An inequality for rearrangements.” Amer. Math. Monthly 60, 176–179.CrossRefGoogle Scholar
Massey, W. A. (1987). “Stochastic ordering for Markov processes on partially ordered spaces.” Math. Operat. Res. 12, 350–367.CrossRefGoogle Scholar
Mira, A. (2001). “Efficiency increasing and stationarity preserving probability mass transfers for MCMC.” Statist. Prob. Lett. 54, 405–411.CrossRefGoogle Scholar
Mira, A. and Geyer, C. J. (1999). “Ordering Monte Carlo Markov chains.” Submitted.Google Scholar
Müller, A. and Stoyan, D. (2002).” Comparison Methods for Stochastic Models and Risks. ” John Wiley, Chichester.Google Scholar
Peskun, P. H. (1973). “Optimum Monte Carlo sampling using Markov chains.” Biometrika 60, 607–612.CrossRefGoogle Scholar
Rüschendorf, L. (1980). “Inequalities for the expectation of δ monotone functions.” Z. Wahrscheinlichkeitsth. 54, 341–349.CrossRefGoogle Scholar
Rüschendorf, L. (2004). “Comparison of multivariate risks and positive dependence.” Adv. Appl. Prob. 41, 391–406.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (1994).” Stochastic Orders and Their Applications. ” Academic Press, Boston, MA.Google Scholar
Szekli, R. (1995).” Stochastic Ordering and Dependence in Applied Probability (Lecture Notes Statist. 97). Springer, New York.CrossRefGoogle Scholar
Szekli, R., Disney, R. L. and Hur, S. (1994). “{MR/GI/1} queues with positively correlated arrival streams.” J. Appl. Prob. 31, 497–514.CrossRefGoogle Scholar
Tchen, A. (1980). “Inequalities for distributions with given marginals.” Ann. Prob. 8, 811–827.CrossRefGoogle Scholar
Tierney, L. (1998). “A note on Metropolis–Hastings kernels for general state spaces.” Ann. Appl. Prob. 8, 1–9.CrossRefGoogle Scholar
Van Doorn, E. (1981).” Stochastic Monotonicity of Birth–Death Processes (Lecture Notes Statist. 4). Springer, Berlin.CrossRefGoogle Scholar
Whitt, W. (1976). “Bivariate distributions with given marginals.” Ann. Statist. 4, 1280–1289.CrossRefGoogle Scholar