Approximations for interactive Markov chains in discrete and continuous time

Abstract

A class of models called interactive Markov chains is studied in both discrete and continuous time. These models were introduced by Conlisk and serve as a rich class for sociological modeling, because they allow for interactions among individuals. In discrete time, it is proved that the Markovian processes converge to a deterministic process almost surely as the population size becomes infinite. More importantly, the normalized process is shown to be asymptotically normal with specified mean vector and covariance matrix. In continuous time, the chain is shown to converge weakly to a diffusion process with specified drift and scale terms. The distributional results will allow for the construction of a likelihood function from interactive Markov chain data, so these results will be important for questions of statistical inference. An example from manpower planning is given which indicates the use of this theory in constructing and evaluating control policies for certain social systems.