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Error bounds for augmented truncation approximations of Markov chains via the perturbation method

Part of: Distribution theory Markov processes

Published online by Cambridge University Press:  26 July 2018

Yuanyuan Liu  and
Wendi Li
Yuanyuan Liu*
Affiliation:
Central South University
Wendi Li*
Affiliation:
Central South University
*
* Postal address: School of Mathematics and Statistics, Central South University, Changsha, Hunan, 410075, China.
* Postal address: School of Mathematics and Statistics, Central South University, Changsha, Hunan, 410075, China.
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Abstract

Let P be the transition matrix of a positive recurrent Markov chain on the integers with invariant probability vector πT, and let (n)P̃ be a stochastic matrix, formed by augmenting the entries of the (n + 1) x (n + 1) northwest corner truncation of P arbitrarily, with invariant probability vector (n)πT. We derive computable V-norm bounds on the error between πT and (n)πT in terms of the perturbation method from three different aspects: the Poisson equation, the residual matrix, and the norm ergodicity coefficient, which we prove to be effective by showing that they converge to 0 as n tends to ∞ under suitable conditions. We illustrate our results through several examples. Comparing our error bounds with the ones of Tweedie (1998), we see that our bounds are more applicable and accurate. Moreover, we also consider possible extensions of our results to continuous-time Markov chains.

Keywords

Markov chainapproximationinvariant probability vectorPoisson equationergodicity

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 62E17: Approximations to distributions (nonasymptotic)
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 2 , June 2018 , pp. 645 - 669
Copyright
Copyright © Applied Probability Trust 2018 

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