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On a fundamental identity in the theory of semi-Markov processes

Published online by Cambridge University Press:  01 July 2016

E. Arjas
E. Arjas*
Affiliation:
The Academy of Finland, Helsinki
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Abstract

A fundamental identity, due to Miller (1961a), (1962a, b) and Kemperman (1961), is generalized to semi-Markov processes. Thus the identity applies to processes defined on a Markov chain with discrete state space and random walks with Markov dependent steps (Section 2). Wald's identity is discussed briefly in Section 3. Section 4 is a study of the maxima of partial sums, and Section 5 of maxima in a semi-Markov process.

Keywords

SEMI-MARKOV KERNELSTOPPING TIMEFUNDAMENTAL IDENTITYWALD'S IDENTITYMAXIMUM OF PARTIAL SUMSMAXIMAL STATE
Type
Research Article
Information
Advances in Applied Probability , Volume 4 , Issue 2 , August 1972 , pp. 258 - 270
Copyright
Copyright © Applied Probability Trust 1972 

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References

Arjas, E. (1972a) On the asymptotic behaviour of a generalization of Markov renewal processes. Soc. Sci. Fenn. Comment. Phys.-Math. 42, 17–25.Google Scholar
Arjas, E. (1972b) On the use of a fundamental identity in the theory of semi-Markov queues. Adv. Appl. Prob. 4, 271–284.Google Scholar
Baxter, G. (1958) An operator identity. Pacific J. Math. 8, 649–663.Google Scholar
Çinlar, E. (1969a) On semi-Markov processes on arbitrary spaces. Proc. Camb. Philos. Soc. 66, 381–392.Google Scholar
Çinlar, E. (1969b) Markov renewal theory. Adv. Appl. Prob. 1, 123–187.Google Scholar
Dinges, H. (1969) Wiener-Hopf-Faktorisierung für substochastische Übergangs-funktionen in angeordneten Räumen. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 11, 152–164.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications. Vol. 2, Second Ed. Wiley, New York.Google Scholar
Kemperman, J. H. B. (1961) The Passage Problem for a Stationary Markov Chain. The University of Chicago Press.Google Scholar
Kingman, J. F. C. (1966) On the algebra of queues. J. Appl. Prob. 3, 285–326.Google Scholar
Matthews, J. P. (1971) A study of processes associated with a finite Markov chain. , University of Sheffield (unpublished).Google Scholar
Miller, H. D. (1961a) A generalization of Wald's identity with applications to random walks. Ann. Math. Statist. 32, 549–560.CrossRefGoogle Scholar
Miller, H. D. (1961b) A convexity property in the theory of random variables defined on a finite Markov chain. Ann. Math. Statist. 32, 1260–1270.Google Scholar
Miller, H. D. (1962a) A matrix factorization problem in the theory of random variables defined on a finite Markov chain. Proc. Camb. Philos. Soc. 58, 268–285.Google Scholar
Miller, H. D. (1962b) Absorption probabilities for sums of random variables defined on a finite Markov chain. Proc. Camb. Philos. Soc. 58, 286–298.Google Scholar
Presman, E. L. (1969) Factorization methods and boundary problems for sums of random variables given on Markov chains. Math. USSR Izv. 3, 815–852. (English translation.)Google Scholar
Spitzer, F. (1956) A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc. 82, 323–339.Google Scholar
Stone, L. D. (1968) On the distribution of the maximum of a semi-Markov process. Ann. Math. Statist. 39, 947–956.Google Scholar
Stone, L. D. (1969) On the distribution of the supremum functional for semi-Markov processes with continuous state space. Ann. Math. Statist. 40, 844–853.Google Scholar
Täcklind, S. (1942) Sur le risque de ruine dans des jeux inéquitables. Skand. Aktuarietidsk. 25, 1–42.Google Scholar