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Asymptotics of locally-interacting Markov chains with global signals

Part of: Special processes Markov 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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