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On determining absorption probabilities for Markov chains in random environments

Published online by Cambridge University Press:  01 July 2016

Richard D. Bourgin  and
Robert Cogburn
Richard D. Bourgin*
Affiliation:
Howard University
Robert Cogburn*
Affiliation:
The University of New Mexico
*
∗Postal address: Department of Mathematics, Howard University, Washington, DC 20059, U.S.A.
∗∗Postal address: Department of Mathematics and Statistics, The University of New Mexico, Albuquerque, NM 87131, U.S.A.
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Abstract

The general framework of a Markov chain in a random environment is presented and the problem of determining extinction probabilities is discussed. An efficient method for determining absorption probabilities and criteria for certain absorption are presented in the case that the environmental process is a two-state Markov chain. These results are then applied to birth and death, queueing and branching chains in random environments.

Keywords

MARKOV CHAINSRANDOM ENVIRONMENTSMARKOVIAN ENVIRONMENTSBICHAINEXTINCTION PROBABILITIESBIRTH AND DEATHQUEUEINGBRANCHING
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 2 , June 1981 , pp. 369 - 387
Copyright
Copyright © Applied Probability Trust 1981 

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References

Cogburn, R. (1980) Markov chains in random environments. Ann. Prob. 8, 908–916.CrossRefGoogle Scholar
Feller, W. (1968) An Introduction to Probability Theory and its Applications, Vol. 1, 3rd ed. Wiley, New York.Google Scholar
Sanchez, E. A. (1978) A Random Walk in a Random Environment. . The University of New Mexico, Albuquerque.Google Scholar
Smith, W. L. and Wilkinson, W. E. (1971) Branching processes in Markovian environment. Duke Math. J. 38, 749–763.Google Scholar
Torrez, W. (1978) The birth and death chain in a random environment: instability and extinction theorems. Ann. Prob. 6, 1026–1043.Google Scholar
Torrez, W. (1979) Calculating extinction probabilities for the birth and death chain in a random environment. J. Appl. Prob. 16, 709–720.Google Scholar
Wilkinson, W. E. (1969) On calculating extinction probabilities for branching processes in random environments. J. Appl. Prob. 6, 478–492.Google Scholar