Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-10T17:31:52.171Z Has data issue: false hasContentIssue false
  • English
  • Français

Geometric and exponential decay in derived Markov chains

Published online by Cambridge University Press:  14 July 2016

Herman Callaert
Herman Callaert*
Affiliation:
Center for Operations Research and Econometrics, University of Leuven
Get access Rights & Permissions [Opens in a new window]

Abstract

It is known [9] that geometric or exponential decay of a Markov chain is preserved under derivation (as defined by Cohen [2]). In this paper we consider the inverse problem, i.e., does a derived Markov chain with a geometrical or exponential decay necessarily arise from a Markov chain having the same property. For a large class of Markov chains (including time reversible chains) a complete solution is found. The method used in this paper proved accurate for obtaining exact results on the value of the decay parameter. An example extending results of Miller [8] and Teugels [9] illustrates the procedure.

Keywords

DERIVED MARKOV CHAINSTRANSITION PROBABILITIESGEOMETRIC AND EXPONENTIAL DECAYSUBORDINATED STOCHASTIC PROCESSES
Type
Short Communications
Information
Journal of Applied Probability , Volume 11 , Issue 2 , June 1974 , pp. 388 - 393
Copyright
Copyright © Applied Probability Trust 1974 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Callaert, H. (1974) On the rate of convergence in birth-and-death processes. Bull. Soc. Math. Belg. To appear.Google Scholar
[2] Cohen, J. W. (1962) Derived Markov chains I, II, III. Nederl. Akad. Wetensch. Proc. Ser. A. 65, 55–92.Google Scholar
[3] Cohen, J. W. (1969) The Single Server Queue. North Holland, Amsterdam.Google Scholar
[4] De Smit, J. H. A. (1972) The asymptotic behaviour of the queue length process in the system M/M/s. CORE Discussion Paper 7217, University of Leuven.Google Scholar
[5] Feller, W. (1966) An Introduction to Probability Theory and its Applications. Vol. II. Wiley, New York.Google Scholar
[6] Kendall, D. G. (1959) Unitary dilations of Markov transition operators and the corresponding integral representations for transition-probability matrices. In Probability and Statistics. (Ed. Grenander, U.) Almqvist and Wiksell, Stockholm.Google Scholar
[7] Kingman, J. F. C. (1963) The exponential decay of Markov transition probabilities. Proc. London Math. Soc. 13, 337–358.Google Scholar
[8] Miller, H. D. (1966) Geometric ergodicity in a class of denumerable Markov chains. Z. Wahrscheinlichkeitsth. 4, 354–373.Google Scholar
[9] Teugels, J. L. (1968) Exponential ergodicity in derived Markov chains. J. Appl. Prob. 5, 669–678.Google Scholar
[10] Vere-Jones, D. (1962) Geometric ergodicity in denumerable Markov chains. Quart. J. Math. Oxford Ser. 13, 7–28.Google Scholar
[11] Widder, D. V. (1946) The Laplace Transform. Princeton University Press, Princeton.Google Scholar