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Entropy of killed-resurrected stationary Markov chains

Part of: Measure-theoretic ergodic theory Markov processes

Published online by Cambridge University Press:  25 February 2021

Servet Martínez
Servet Martínez*
Affiliation:
Universidad de Chile
*
*Postal address: Departamento Ingeniería Matemática and Centro Modelamiento Matemático, Universidad de Chile, UMI 2807 CNRS, Casilla 170-3, Correo 3, Santiago, Chile. Email address: [email protected]
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Abstract

We consider a strictly substochastic matrix or a stochastic matrix with absorbing states. By using quasi-stationary distributions we show that there is an associated canonical Markov chain that is built from the resurrected chain, the absorbing states, and the hitting times, together with a random walk on the absorbing states, which is necessary for achieving time stationarity. Based upon the 2-stringing representation of the resurrected chain, we supply a stationary representation of the killed and the absorbed chains. The entropies of these representations have a clear meaning when one identifies the probability measure of natural factors. The balance between the entropies of these representations and the entropy of the canonical chain serves to check the correctness of the whole construction.

Keywords

Markov chainsentropyquasi-stationary distributions

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 28D20: Entropy and other invariants
Type
Research Papers
Information
Journal of Applied Probability , Volume 58 , Issue 1 , March 2021 , pp. 177 - 196
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Abramov, L. and Rokhlin, V. (1962). The entropy of a skew product of measure-preserving transformations. Vestnik Leningrad Univ. 17, 5–13 (in Russian).Google Scholar
Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. New York, Springer.Google Scholar
Collet, P., Martínez, S. and San Martín, J. (2013). Quasi-Stationary Distributions: Markov Chains, Diffusions and Dynamical Systems. Heidelberg, Springer.CrossRefGoogle Scholar
Darroch, J. and Seneta, E. (1965). On quasi-stationary distributions in absorbing discrete-time finite Markov chains. J. Appl. Prob. 2, 88–100.CrossRefGoogle Scholar
Denker, M., Grillenberger, C. and Sigmund, K. (1976). Ergodic Theory on Compact Spaces. Berlin, Springer.CrossRefGoogle Scholar
Downarowicz, T. and Serafin, J. (2002). Fiber entropy and conditional variational principles in compact non-metrizable spaces. Fund. Math. 172, 217–247.CrossRefGoogle Scholar
Ferrari, P. A., Kesten, H., Martínez, S. and Picco, P. (1995). Existence of quasi-stationary distributions. A renewal dynamical approach. Ann. Prob. 23, 501–521.CrossRefGoogle Scholar
Ferrari, P. A., Martínez, S. and Picco, P. (1992). Existence of non-trivial quasi-stationary distributions in the birth-death chain. Adv. Appl. Prob. 24, 795–813.CrossRefGoogle Scholar
Keane, M. and Smorodinsky, M. (1979). Finitary isomorphisms of irreducible Markov shifts. Israel J. Math. 34, 281–286.CrossRefGoogle Scholar