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Markov chains in small time intervals

Published online by Cambridge University Press:  14 July 2016

Stig I. Rosenlund
Stig I. Rosenlund*
Affiliation:
University of Göteborg
*
∗Postal address: Västmannagatan 93, S-113 43 Stockholm, Sweden.
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Abstract

For a time-homogeneous continuous-parameter Markov chain we show that as t → 0 the transition probability pn,j (t) is at least of order where r(n, j) is the minimum number of jumps needed for the chain to pass from n to j. If the intensities of passage are bounded over the set of states which can be reached from n via fewer than r(n, j) jumps, this is the exact order.

Keywords

BIRTH-DEATH PROCESSLOCAL BEHAVIOURMARKOV CHAIN
Type
Short Communications
Information
Journal of Applied Probability , Volume 18 , Issue 3 , September 1981 , pp. 747 - 751
Copyright
Copyright © Applied Probability Trust 

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References

Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. Wiley, New York.Google Scholar
Rosenlund, S. I. (1978a) Transition probabilities for a truncated birth-death process. Scand. J. Statist. 5, 119–122.Google Scholar
Rosenlund, S. I. (1978b) Controlling a Markovian queue at equidistant time epochs. Trans. 8th Prague Conf. Information Theory, Statistical Decision Functions, Random Processes, Vol. B, ed. Driml, M. Academia, Prague, 129–143.Google Scholar