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Discrete-time singularly perturbed Markov chains: aggregation, occupation measures, and switching diffusion limit

Part of: Asymptotic theory Markov processes Limit theorems Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  22 February 2016

G. Yin ,
Q. Zhang  and
G. Badowski
G. Yin*
Affiliation:
Wayne State University
Q. Zhang*
Affiliation:
University of Georgia
G. Badowski*
Affiliation:
Wayne State University
*
∗ Postal address: Department of Mathematics, Wayne State University, Detroit, MI 48202, USA. Email address: gyin@math.wayne.edu
∗∗ Postal address: Department of Mathematics, University of Georgia, Athens, GA 30602, USA.
∗∗∗ Current address: Institute for Systems Research, University of Maryland, College Park, MD 20742, USA.
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Abstract

This work is devoted to asymptotic properties of singularly perturbed Markov chains in discrete time. The motivation stems from applications in discrete-time control and optimization problems, manufacturing and production planning, stochastic networks, and communication systems, in which finite-state Markov chains are used to model large-scale and complex systems. To reduce the complexity of the underlying system, the states in each recurrent class are aggregated into a single state. Although the aggregated process may not be Markovian, its continuous-time interpolation converges to a continuous-time Markov chain whose generator is a function determined by the invariant measures of the recurrent states. Sequences of occupation measures are defined. A mean square estimate on a sequence of unscaled occupation measures is obtained. Furthermore, it is proved that a suitably scaled sequence of occupation measures converges to a switching diffusion.

Keywords

Markov chaindiscrete timesingular perturbationoccupation measureswitching diffusionaggregationweak convergence

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60B10: Convergence of probability measures 60F17: Functional limit theorems; invariance principles 34E05: Asymptotic expansions
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 2 , June 2003 , pp. 449 - 476
Copyright
Copyright © Applied Probability Trust 2003 

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Footnotes

*

Supported in part by the National Science Foundation under grants DMS-9877090.

**

Supported in part by the USAF Grant F30602-99-2-0548 and ONR Grant N00014-96-1-0263.

***

Supported in part by Wayne State University.

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