Convergence of locally and globally interacting Markov chains.
The asymptotic behaviour of interactive Markov chains on infinite product spaces is studied. With a fixed finite state space \(C\) the state space of the process is therefore obtained as a set of configurations of the individual states, the index set being given by an infinite set on sites or agents \(A\). Typically, the dynamics of such process is described by a transition kernel which assumes a product form. Each term of the product expresses the probability that the corresponding agent switches to some particular state. This probability usually does not depend only on the present state of the agent but also on the states of some other agents. The convergence behaviour of such Markov chains has been investigated in depth in the case where the interaction is purely local, i.e.\ if the transition probability for each agent depends only on the states of the agents in some finite neighborhood. In the present paper, motivated by analysing financial markets, a global component in the interaction is introduced, in particular the transition law is assumed to depend on the average behaviour of the whole configuration. These can be viewed as temporary price equilibria for the evolution of the stock price. The authors give sufficient conditions for convergence of such Markov chains on macroscopic level of empirical fields. Using a perturbation of the Dobrushin-Vasserstein contraction technique they show that macroscopic convergence implies weak convergence of the underlying Markov chain. This extends the basic convergence of L. N. Vasserstein for locally interacting Markov chains to the case where additional global component appears in the interaction.
Markov chains on infinite product spaces
scientific article; zbMATH DE number 2138905