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Asymptotics of locally-interacting Markov chains with global signals

Part of: Markov processes Special processes

Published online by Cambridge University Press:  01 July 2016

Ulrich Horst
Ulrich Horst*
Affiliation:
Humboldt-Universität zu Berlin
*
∗ Current address: Bendheim Center for Finance, Dial Lodge, 26 Prospect Avenue, Princeton University, Princeton, NJ 08540-5296, USA. Email address: uhorst@princeton.edu
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Abstract

We study the long-run behaviour of interactive Markov chains on infinite product spaces. The behaviour at a single site is influenced by the local situation in some neighbourhood and by a random signal about the average situation throughout the whole system. The asymptotic behaviour of such Markov chains is analyzed on the microscopic level and on the macroscopic level of empirical fields. We give sufficient conditions for convergence on the macroscopic level. Combining a convergence result from the theory of random systems with complete connections with a perturbation of the Dobrushin-Vasserstein contraction technique, we show that macroscopic convergence implies that the underlying microscopic process has local asymptotic loss of memory.

Keywords

Markov chains on infinite product spaceslocal asymptotic loss of memoryGibbs measuresbehavioural financerandom systems with complete connections

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 34 , Issue 2 , June 2002 , pp. 416 - 440
Copyright
Copyright © Applied Probability Trust 2002 

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References

Brock, W. A. and Hommes, C. (1997). A rational route to randomness. Econometrica 65, 1059–1095.CrossRefGoogle Scholar
Dobrushin, R. L. (1968). Description of random fields by means of conditional probabilities and the conditions of its regularity. Theory Prob. Appl. 13, 201–229.Google Scholar
Föllmer, H., (1979a). Macroscopic convergence of Markov chains on infinite product spaces. In Random Fields (Colloq. Math. Soc. Janos Bolyai 27), North-Holland, Amsterdam, pp. 363–371.Google Scholar
Föllmer, H., (1979b). Tail structure of Markov chains on infinite product spaces. Z. Wahrscheinlichkeitsth. 50, 273–285.CrossRefGoogle Scholar
Föllmer, H., (1982). A covariance estimate for Gibbs measures. J. Funct. Anal. 46, 387–395.Google Scholar
Föllmer, H. and Horst, U. (2001). Convergence of locally and globally interacting Markov chains. Stoch. Process. Appl. 96, 99–121.Google Scholar
Georgii, H. (1988). Gibbs Measures and Phase Transitions. Walter de Gruyter, Berlin.CrossRefGoogle Scholar
Horst, U. (2000). Asymptotics of locally and globally interacting Markov chains arising in microstructure models of financial markets. , Humboldt-Universität zu Berlin.Google Scholar
Iosifescu, M. and Grigorescu, S. (1990). Dependence with Complete Connections and Its Applications. Cambridge University Press.Google Scholar
Iosifescu, M. and Theodorescu, R. (1968). Random Processes and Learning. Springer, Berlin.Google Scholar
Kirman, A. (1998). On the transitory nature of gurus. Working paper, EHESS and Université de Marseille III.Google Scholar
Lebowitz, J. L., Maes, C. and Speer, E. R. (1990). Statistical mechanics of probabilistic cellular automata. J. Statist. Phys. 59, 117–170.CrossRefGoogle Scholar
Norman, F. M. (1972). Markov Processes and Learning Models. Academic Press, New York.Google Scholar
Simon, B. (1993). The Statistical Mechanics of Lattice Gases. Princeton University Press.Google Scholar
Vasserstein, L. N. (1969). Markov processes over denumerable product of spaces describing large systems of automata. Problemy Peredaci Informacii 5, 64–72.Google Scholar