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Moments of age and remaining life in renewal processes

Published online by Cambridge University Press:  14 July 2016

S. Katz ,
P. Naor  and
R. Shinnar
S. Katz
Affiliation:
The City College of The City University of New York
P. Naor
Affiliation:
The City College of The City University of New York
R. Shinnar
Affiliation:
The City College of The City University of New York
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Abstract

Expressions are developed for the moments of age and remaining life in renewal processes, and from them, expressions for the moments of the last and next renewal epochs. The results for remaining life and next renewal epoch may be regarded as generalizations of theorems of Wald (1944) and others. The results for age and last renewal epoch do not fit into this sequential analysis framework. Both sets of results are obtained from partial differential equations developed for the distributions of age and of remaining life. These partial differential equations are of the “conservation” type familiar from statistical mechanics, and may have some independent interest for renewal theory.

Keywords

RENEWAL PROCESSESAGE DISTRIBUTIONSREMAINING LIFETIME DISTRIBUTIONSOJOURN TIMESRENEWAL EPOCH
Type
Research Papers
Information
Journal of Applied Probability , Volume 10 , Issue 2 , June 1973 , pp. 307 - 316
Copyright
Copyright © Applied Probability Trust 1973 

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References

Cox, D. R. (1962) Renewal Theory. Methuen, London. Pp. 61–63.Google Scholar
Hulburt, H. M. and Katz, S. (1964) Some problems in particle technology: a statistical mechanical formulation. Chem. Eng. Sci. 19, 555–574.CrossRefGoogle Scholar
Johnson, N. L. (1959) A proof of Wald's theorem on cumulative sums. Ann. Math. Statist. 30, 1245–1247.Google Scholar
Kolmogorov, A. N. and Prokhorov, Yu. V. (1949) O summakh sluchaynovo chisla sluchaynikh slagaemikh. Uspekhi Matem. Nauk (N.S.) IV, 4, 168–172.Google Scholar
Robbins, H. (1970) Optimal stopping. Amer. Math. Monthly 77, 333–343.CrossRefGoogle Scholar
Wald, A. (1944) On cumulative sums of random variables. Ann. Math. Statist. 15, 283–296.Google Scholar
Wolfowitz, J. (1947) The efficiency of sequential estimates and Wald's equation for sequential processes. Ann. Math. Statist. 18, 215–230.CrossRefGoogle Scholar