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A cluster process representation of a self-exciting process

  • Alan G. Hawkes (a1) and David Oakes (a2)
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.

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Corresponding author
* Now at University College Swarsea.
** Now at Harvard University.
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Supported by the Science Research Council and N. S. F. grant GS32327X.

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References
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[1] Bartlett, M. S. (1955) An Introduction to Stochastic Processes. Cambridge University Press.
[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, 6567.
[3] Cox, D. R. and Lewis, P. A. W. (1971) Multivariate point processes. Proc. Sixth. Berk. Symp. on Math. Statist. and Prob.
[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.
[5] Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.
[6] Hawkes, A. G. (1971a) Spectra of some self-exciting and mutually exciting point processes. Biometrika 58, 8390.
[7] Hawkes, A. G. (1971b) Point spectra of some mutually exciting point processes. J. R. Statist. Soc. B 33, 438443.
[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.
[9] Kendall, D. G. (1949) Stochastic processes and population growth. J. R. Statist. Soc. B 11, 230264.
[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 399415. Czechoslovakia Academy of Sciences, Prague.
[11] Lewis, P. A. W. (1964) A branching Poisson process model for the analysis of computer failure patterns. J. R. Statist. Soc. B 26, 398456.
[12] Lewis, P. A. W. (1969) Asymptotic properties and equilibrium conditions for branching Poisson processes. J. Appl. Prob. 6. 355371.
[13] Vere-Jones, D. (1970) Stochastic models for earthquake occurrence. J. R. Statist. Soc. B 32, 162.
[14] Westcott, M. (1971) On existence and mixing results for cluster point processes. J. R. Statist. Soc. B 33, 290300.
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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
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