Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T18:43:02.352Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
∗∗ 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: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Aldous, D. and Diaconis, P. (1987). Strong uniform times and finite random walks. Adv. Appl. Math. 8, 69–97.Google Scholar
Carmona, P., Petit, F. and Yor, M. (1998). Beta-gamma random variables and interwining relations between certain Markov processes. Rev. Mat. Iberoamericana 14, 311–367.CrossRefGoogle Scholar
Dellacherie, C., Martínez, S. and San Martín, J. (2000). Description of the sub-Markov kernel associated to generalized ultrametric matrices. An algorithmic approach. Linear Algebra Appl. 318, 1–21.Google Scholar
Diaconis, P. and Fill, J. A. (1990). Strong stationary times via a new form of duality. Ann. Prob. 18, 1483–1522.CrossRefGoogle Scholar
Fill, J. A. (2009). The passage time distribution for a birth-and-death chain: strong stationary duality gives a first stochastic proof. J. Theoret. Prob. 22, 543–557.Google Scholar
Fill, J. A. (2009). On hitting times and fastest strong stationary times for skip-free and more general chains. J. Theoret. Prob. 22, 587–600.CrossRefGoogle Scholar
Horn, R. A. and Johnson, C. R. (1985). Matrix Analysis. Cambridge University Press.Google Scholar
Huillet, T. E. (2009). A duality approach to the genealogies of discrete non-neutral Wright–Fisher models. J. Prob. Statist. 2009, 22pp.Google Scholar
Huillet, T. E. and Möhle, M. (2009). Duality and asymptotics for a class of nonneutral discrete Moran models. J. Appl. Prob. 46, 866–893.CrossRefGoogle Scholar
Keilson, J. (1979). Markov Chain Models—Rarity and Exponentiality (Appl. Math. Sci. 28). Springer, New York.Google Scholar
Liggett, T. M. (1985). Interacting Particle Systems (Fund. Principles Math. Sci. 276). Springer, New York.Google Scholar
Martínez, S., Michon, G. and San Martín, J. (1994). Inverse of ultrametric matrices are of Stieltjes type. SIAM J. Matrix Anal. Appl. 15, 98–106.Google Scholar
McDonald, J. J., Neumann, M., Schneider, H. and Tsatsomeros, M. J. (1995). Inverse M-matrix inequalities and generalized ultrametric matrices. Linear Algebra Appl. 220, 321–341.Google Scholar
Möhle, M. (1999). The concept of duality and applications to Markov processes arising in neutral population genetics models. Bernoulli 5, 761–777.Google Scholar
Nabben, R. and Varga, R. S. (1995). Generalized ultrametric matrices—a class of inverse M-matrices. Linear Algebra Appl. 220, 365–390.Google Scholar
Siegmund, D. (1976). The equivalence of absorbing and reflecting barrier problems for stochastically monotone Markov processes. Ann. Prob. 4, 914–924.Google Scholar
Sudbury, A. and Lloyd, P. (1995). Quantum operators in classical probability theory. II. The concept of duality in interacting particle systems. Ann. Prob. 23, 1816–1830.Google Scholar