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On the moments of Markov renewal processes

Published online by Cambridge University Press:  01 July 2016

Jeffrey J. Hunter
Jeffrey J. Hunter*
Affiliation:
University of North Carolina
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Abstract

Recently Kshirsagar and Gupta [5] obtained expressions for the asymptotic values of the first two moments of a Markov renewal process. The method they employed involved formal inversion of matrices of Laplace-Stieltjes transforms. Their method also required the imposition of a non-singularity condition. In this paper we derive the asymptotic values using known renewal theoretic results. This method of approach utilises the fundamental matrix of the imbedded ergodic Markov chain and the theory of generalised matrix inverses. Although our results differ in form from those obtained by Kshirsagar and Gupta [5] we show that they reduce to their results under the added non-singularity condition. As a by-product of the derivation we find explicit expressions for the moments of the first passage time distributions in the associated semi-Markov process, generalising the results of Kemeny and Snell [4] obtained for Markov chains.

Type
Research Article
Information
Advances in Applied Probability , Volume 1 , Issue 2 , Autumn 1969 , pp. 188 - 210
Copyright
Copyright © Applied Probability Trust 

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References

[1] Barlow, R. E. and Proschan, F. (1965) Mathematical Theory of Reliability. Wiley, New York.Google Scholar
[2] Fréchet, M. (1950) Recherches Théoriques Modernes sur le Calcul des Probabilités (Vol. 2, Second Edition). Gauthier-Villars, Paris.Google Scholar
[3] Keilson, J. and Wishart, D. M. G. (1964) A central limit theorem for processes defined on a finite Markov chain. Proc. Camb. Phil. Soc. 60, 547–567.Google Scholar
[4] Kemeny, J. G. and Snell, J. L. (1960) Finite Markov Chains. Van Nostrand, Princeton, N. J.Google Scholar
[5] Kshirsagar, A. M. and Gupta, Y. P. (1967) Asymptotic values of the first two moments in Markov renewal processes. Biometrika 54, 597–603.CrossRefGoogle ScholarPubMed
[6] Murthy, V. K. (1961) On the general renewal process. Institute of Statistics Mimeo Series No. 293, University of North Carolina.Google Scholar
[7] Pyke, R. (1961) Markov renewal processes: Definitions and preliminary properties. Ann. Math. Statist. 32, 1231–1242.CrossRefGoogle Scholar
[8] Pyke, R. (1961) Markov renewal processes with finitely many states. Ann. Math. Statist. 32, 1243–1259.CrossRefGoogle Scholar
[9] Rao, C. R. (1965) Linear Statistical Inference and its Applications, Wiley, New York.Google Scholar
[10] Rao, C. R. (1966) Generalised inverse for matrices and its applications in mathematical statistics. Research Papers in Statistics, Festschrift for J. Neyman. Wiley, New York.Google Scholar
[11] Smith, W. L. (1955) Regenerative stochastic processes. Proc. Roy. Soc. A 232, 6–31.Google Scholar
[12] Smith, W. L. (1958) Renewal theory and its ramifications. J. R. Statist. Soc. B 20, 243–302.Google Scholar
[13] Smith, W. L. (1959) On the cumulants of renewal processes. Biometrika 46, 1–29.Google Scholar