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Stochastic Systems: Modeling, Identification and Optimization, I pp 67–78Cite as

Limit theorems and diffusion approximations for density dependent Markov chains

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Part of the book series: Mathematical Programming Studies ((MATHPROGRAMM,volume 5))

Abstract

One parameter families of Markov chains X A (t) with infinitesimal parameters given by q A k,k+l =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, \(\dot X = F(X)\)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.

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Authors and Affiliations

  1. University of Wisconsin, Madison, Wis., U.S.A.

    Thomas G. Kurtz

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  1. Thomas G. Kurtz
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Roger J.- B. Wets

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© 1976 The Mathematical Programming Society

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Kurtz, T.G. (1976). Limit theorems and diffusion approximations for density dependent Markov chains. In: Wets, R.J.B. (eds) Stochastic Systems: Modeling, Identification and Optimization, I. Mathematical Programming Studies, vol 5. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0120765

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  • DOI: https://doi.org/10.1007/BFb0120765

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00783-5

  • Online ISBN: 978-3-642-00784-2

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