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Duality and intertwining for discrete Markov kernels: relations and examples

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Thierry Huillet  and
Servet Martinez
Thierry Huillet*
Affiliation:
Université de Cergy-Pontoise
Servet Martinez*
Affiliation:
Universidad de Chile
*
∗ Postal address: Laboratoire de Physique Théorique et Modélisation, CNRS UMR 8089, Université de Cergy-Pontoise, 2 Avenue Adolphe Chauvin, 95302 Cergy-Pontoise, France. Email address: thierry.huillet@u-cergy.fr
∗∗ Postal address: Departamento de Ingeniería Matemática and Centro Modelamiento Matemático (CNRS UMI 2807), Universidad de Chile, Casilla 170/3, Correo 3, Santiago, Chile. Email address: smartine@dim.uchile.cl
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Abstract

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We supply some relations that establish intertwining from duality and give a probabilistic interpretation. This is carried out in the context of discrete Markov chains, fixing up the background of previous relations established for monotone chains and their Siegmund duals. We revisit the duality for birth-and-death chains and the nonneutral Moran model, and we also explore the duality relations in an ultrametric-type dual that extends the Siegmund kernel. Finally, we discuss the sharp dual, following closely the Diaconis-Fill study.

Keywords

DualityintertwiningSiegmund dualgeneralized ultrametric matrixbirth-and-death chainMoran modelsharp strong stationary timesharp dual

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 2 , June 2011 , pp. 437 - 460
Copyright
Copyright © Applied Probability Trust 2011 

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