Limit theorems and diffusion approximations for density dependent Markov chains

Abstract

One parameter families of Markov chains X A (t) with infinitesimal parameters given by q k,k+l A =Af(A −1 k,l) k, l ∈Z′ l≠0 are considered. Under appropriate conditions X A (t)/A converges in probability as A→∞ to a solution of the system of ordinary differential equations, X˙=F(X) where F(x)=σt lf(x, l). Limit theorems for these families are reviewed including work of Norman, Barbour and the author. A natural diffusion approximation is discussed. Families of this type include the usual epidemic model, models in chemistry, genetics and in many other areas of application.