Convergence of locally and globally interacting Markov chains

We study the long run behaviour of interactive Markov chains on infinite pr oduct spaces. In view of microstructure models of financial markets, the in teraction has both a local and a global component. The convergence of such Markov chains is analyzed on the microscopic level and on the macroscopic l evel of empirical fields. We give sufficient conditions for convergence on the macroscopic level. Using a perturbation of the Dobrushin-Vasserstein co ntraction technique we show that macroscopic convergence implies weak conve rgence of the underlying Markov chain. This extends the basic convergence t heorem of Vasserstein for locally interacting Markov chains to the case whe re an additional global component appears in the interaction. (C) 2001 Else vier Science B.V. All rights reserved.