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Time series formed from the superposition of discrete renewal processes

Published online by Cambridge University Press:  14 July 2016

P. A. Blight
P. A. Blight*
Affiliation:
Hatfield Polytechnic
*
∗Postal address: School of Information Science, Hatfield Polytechnic, Hatfield, Herts AL10 9AB, UK.
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Abstract

The superposition of independent, discrete, renewal processes produces a counting process which is also a discrete time series. The conditional distribution and correlation structure of this kind of time series may be obtained. In suitable conditions the conditional distribution has a spectrum which is exactly or approximately rational. When this is so, an ARMA can be found which matches the spectrum of the superposition.

Keywords

ARMA TIME SERIESBOX-JENKINS
Type
Short Communications
Information
Journal of Applied Probability , Volume 26 , Issue 1 , March 1989 , pp. 189 - 195
Copyright
Copyright © Applied Probability Trust 1989 

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References

Blight, P. A. (1986) Inventory renewal time series and their ARMA equivalents. Ph.D. Dissertation, Department of Statistics, Birkbeck College, University of London.Google Scholar
Box, G. E. P. and Jenkins, G. M. (1970) Time series, Forecasing and Control. Holden Day, San Francisco.Google Scholar
Cox, D. R. (1962) Renewal Theory. Methuen, London.Google Scholar
Cox, D. R. and Smith, W. L. (1954) On the superposition of renewal processes. Biometrika 41, 91–99.Google Scholar
Erdös, P., Feller, W. and Pollard, H. (1949) A theorem on power series. Bull. Amer. Math. Soc. 55, 201–204.Google Scholar
Feller, W. (1957) An Introduction to Probability Theory and its Applications , 3rd edn. Wiley, New York.Google Scholar
Lawrance, A. J. and Lewis, P. A. W. (1985) Modelling and residual analysis of non-linear autoregressive time series in exponential variables (with discussion). J. R. Statist. Soc. B47, 165–202.Google Scholar
Mckenzie, E. (1985) Some simple models for discrete variate time series. Water Resources Bull. 21(4), 645–650.Google Scholar
Priestley, M. B. (1981) Spectral Analysis and Time Series. Academic Press, London.Google Scholar
Smith, W. L. (1958) Renewal theory and its ramifications. J. R. Statist. Soc. B20, 284–302.Google Scholar