Incollection,
Limit theorems and diffusion approximations for density dependent Markov chains
Stochastic Systems: Modeling, Identification and Optimization, I, volume 5 of Mathematical Programming Studies, Springer Berlin Heidelberg, (1976)
DOI: 10.1007/BFb0120765
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.
MR
The author gives a concise review of the diffusion approximation theory for density dependent Markov processes. Under weak conditions, a law of large numbers holds, in which the trajectory of the process, suitably normalised, converges in probability to that of the deterministic approximation. Analogues of the central limit theorem are also stated, and the paper concludes with the proof of an approximation theorem derived from that of J. Komlós, P. Major and G. Tusnády Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 32 (1975), 111–131; MR 52 #11605b, which gives a precise estimate for the discrepancy between the process and its diffusion approximation. Examples of density dependent processes outlined include epidemic, competition, and chemical reaction models.
For the entire collection see MR0426080 (54 #14026).
