Singularly perturbed Markov chains: Convergence and aggregation
type: Item
instance of: scholarly article
title: Singularly perturbed Markov chains: Convergence and aggregation
zbMATH Open document ID: 0951.60034
author: Q. Zhang
author: G. Badowski
author: G. Yin
published in: Journal of Multivariate Analysis
publication date: 2000-12-18T00:00:00Z
review text: This work develops asymptotic results for singularly perturbed Markov chains. The attention is concentrated on the solution of forward equations, related occupation measures and aggregated processes. The increasing demands on modelling and control of dynamical systems have been made for treating singularly perturbed Markov chains (see V. V. Anisimov (1995), S. P. Sethi and Q. Zhang (1994) among others). For many applications such as communication systems and queueing networks it is difficult to obtain analytic solutions. Direct numerical approximations and straightforward uses of numerical packages may not be possible due to the dimensionality. As an alternative, one decomposes a large dimensional job in small pieces [see \textit{H. A. Simon} and \textit{A. Ando}, Econometrica 29, 111-138 (1961; Zbl 0121.15103)]. Using a singular perturbation approach one separates the states in accordance with their rates of changes. Nonhomogeneous Markov chains and singularly perturbed systems arise in the framework of simulated annealing and global optimization [see \textit{T.-S. Chiang} and \textit{Y. Chow}, Ann. Probab. 17, No. 4, 1483-1502 (1989; Zbl 0687.60070)]. The authors answer the following questions: What can be done for generators that are not very smooth? Can pointwise convergence and convergence in \(L^2\) under the norm topology be obtained under acceptable weak conditions? It should be mentioned that two-time scale approaches and singularly perturbed stochastic systems have also been considered by S. N. Ethier and T. G. Kurtz (1996), H. J. Kushner (1990) and others with the main concerns on diffusion processes, jump diffusions, or systems involving wide-band noise. For ease of presentation, the authors use the transient case as a prototype in Sections 2-4. Section 2 contains the convergence result under the weak topology of \(L^2[0,T]\). Section 3 is dedicated to the study of pointwise convergence and convergence under the \(L^2\) norm topology. Section 4 treates t…
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 34E05
zbMATH DE Number: 1437466
zbMATH Keywords: Markov chains
zbMATH Keywords: simulated annealing
zbMATH Keywords: convergence
MaRDI profile type: Publication
full work available at URL: jmva.1999.1855
OpenAlex ID: W2049581375
cites work: Switching processes: Averaging principle, diffusion approximation and applications
cites work: A limit theorem for a class of inhomogeneous Markov processes
cites work: Q5524074
cites work: Q4144754
cites work: Q3136505
cites work: Optimal control of Markov chains admitting strong and weak interactions
cites work: Q3721531
cites work: Q3672830
cites work: Q5580482
cites work: On the geometrical convergence of Gibbs sampler in \(\mathbb R^d\)
cites work: Q3876725
cites work: Q5639096
cites work: Constructing asymptotic series for probability distributions of Markov chains with weak and strong interactions
cites work: Q3344923
cites work: Weak convergence methods and singularly perturbed stochastic control and filtering problems
cites work: Q4861696
cites work: Q4040350
cites work: A singular perturbation approach to modeling and control of Markov chains
cites work: Q4265503
cites work: Aggregation of Variables in Dynamic Systems
cites work: Q4375487
cites work: Asymptotic properties of a singularly perturbed Markov chain with inclusion of transient states.
DOI: 10.1006/JMVA.1999.1855
description: scientific article; zbMATH DE number 1437466