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A unified stability theory for classical and monotone Markov chains

Part of: Spaces with richer structures Ordered sets Markov processes

Published online by Cambridge University Press:  12 July 2019

Takashi Kamihigashi  and
John Stachurski
Takashi Kamihigashi*
Affiliation:
Kobe University
John Stachurski*
Affiliation:
Australian National University
*
*Postal address: Research Institute of Economics and Business, Kobe University, Japan. Email address: tkamihig@rieb.kobe-u.ac.jp
**Postal address: Research School of Economics, Australian National University, Australia. Email address: john.stachurski@anu.edu.au
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Abstract

In this paper we integrate two strands of the literature on stability of general state Markov chains: conventional, total-variation-based results and more recent order-theoretic results. First we introduce a complete metric over Borel probability measures based on ‘partial’ stochastic dominance. We then show that many conventional results framed in the setting of total variation distance have natural generalizations to the partially ordered setting when this metric is adopted.

Keywords

Total variationMarkov chainstochastic dominationcoupling

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces 60J99: None of the above, but in this section
Secondary: 54E50: Complete metric spaces 06A06: Partial order, general
Type
Research Papers
Information
Journal of Applied Probability , Volume 56 , Issue 1 , March 2019 , pp. 1 - 22
Copyright
© Applied Probability Trust 2019 

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