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A cluster process representation of a self-exciting process

Published online by Cambridge University Press:  14 July 2016

Alan G. Hawkes  and
David Oakes
Alan G. Hawkes*
Affiliation:
University of Durham
David Oakes*
Affiliation:
Imperial College London
*
*Now at University College Swarsea.
**Now at Harvard University.
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Abstract

It is shown that all stationary self-exciting point processes with finite intensity may be represented as Poisson cluster processes which are age-dependent immigration-birth processes, and their existence is established. This result is used to derive some counting and interval properties of these processes using the probability generating functional.

Keywords

CLUSTER PROCESSPOINT PROCESSSELF-EXCITING PROCESS
Type
Research Papers
Information
Journal of Applied Probability , Volume 11 , Issue 3 , September 1974 , pp. 493 - 503
Copyright
Copyright © Applied Probability Trust 1974 

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Footnotes

Supported by the Science Research Council and N. S. F. grant GS32327X.

References

[1] Bartlett, M. S. (1955) An Introduction to Stochastic Processes. Cambridge University Press.Google Scholar
[2] Bartlett, M. S. and Kendall, D. G. (1951) On the use of the characteristic functional in the analysis of some stochastic processes occurring in physics and biology. Proc. Camb. Phil. Soc. 47, 65–67.Google Scholar
[3] Cox, D. R. and Lewis, P. A. W. (1971) Multivariate point processes. Proc. Sixth. Berk. Symp. on Math. Statist. and Prob. Google Scholar
[4] Daley, D. J. and Vere-Jones, D. (1971) A summary of the theory of point processes. Paper presented to the stochastic point process conference, New York, 1971.Google Scholar
[5] Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.Google Scholar
[6] Hawkes, A. G. (1971a) Spectra of some self-exciting and mutually exciting point processes. Biometrika 58, 83–90.Google Scholar
[7] Hawkes, A. G. (1971b) Point spectra of some mutually exciting point processes. J. R. Statist. Soc. B 33, 438–443.Google Scholar
[8] Jowett, J. and Vere-Jones, D. (1971) The prediction of stationary point processes. Paper presented to the stochastic point process conference, New York, 1971.Google Scholar
[9] Kendall, D. G. (1949) Stochastic processes and population growth. J. R. Statist. Soc. B 11, 230–264.Google Scholar
[10] Kerstan, J. and Matthes, K. (1965) Ergodische unbegrenzt teilbare stationare zufallige Punktfolgen. Trans. Fourth Prague Conference on Information Theory, Statistical Decision Functions and Random Processes 399–415. Czechoslovakia Academy of Sciences, Prague.Google Scholar
[11] Lewis, P. A. W. (1964) A branching Poisson process model for the analysis of computer failure patterns. J. R. Statist. Soc. B 26, 398–456.Google Scholar
[12] Lewis, P. A. W. (1969) Asymptotic properties and equilibrium conditions for branching Poisson processes. J. Appl. Prob. 6. 355–371.Google Scholar
[13] Vere-Jones, D. (1970) Stochastic models for earthquake occurrence. J. R. Statist. Soc. B 32, 1–62.Google Scholar
[14] Westcott, M. (1971) On existence and mixing results for cluster point processes. J. R. Statist. Soc. B 33, 290–300.Google Scholar