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A simple proof of the multivariate random time change theorem for point processes

Published online by Cambridge University Press:  14 July 2016

Timothy C. Brown  and
M. Gopalan Nair
Timothy C. Brown*
Affiliation:
University of Melbourne
M. Gopalan Nair
Affiliation:
University of Melbourne
*
∗ Work completed during the tenure of a U.K. Science and Engineering Research Council Visiting Fellowship at the University of Bath, whose hospitality is gratefully acknowledged, and also partly supported by the University of Western Australia.
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Abstract

A simple proof of the multivariate random time change theorem of Meyer (1971) is given. This result includes Watanabe's (1964) characterization of the Poisson process; even in this special case the present proof is simpler than existing proofs.

Keywords

POISSON PROCESSMARTINGALECOMPENSATOR
Type
Short Communications
Information
Journal of Applied Probability , Volume 25 , Issue 1 , March 1988 , pp. 210 - 214
Copyright
Copyright © Applied Probability Trust 1988 

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Footnotes

Present address for both authors: Department of Mathematics, University of Western Australia, Nedlands, WA 6009, Australia.

References

Aalen, O. O. and Hoem, J. M. (1978) Random time changes for multivariate counting processes, Scand. Actuar. J. , 81–101.CrossRefGoogle Scholar
Bremaud, P. (1975) An extension of Watanabe's theorem of characterization of Poisson processes. J. Appl. Prob. 12, 396–399.CrossRefGoogle Scholar
Bremaud, P. (1981) Point Processes and Queues, Martingale Dynamics. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Brown, T. C. (1981) Compensators and Cox convergence. Math. Proc. Camb. Phil. Soc. 90, 305–319.CrossRefGoogle Scholar
Brown, T. C. (1983) Some Poisson approximations using compensators. Ann. Prob. 11, 726–744.CrossRefGoogle Scholar
Dellacherie, C. and Meyer, P. A. (1982) Probabilities and Potential B , North-Holland, Amsterdam.Google Scholar
Jacod, J. (1975) Multivariate point processes: predictable projection, Radon–Nikodym derivatives, representation of martingales. Z. Wahrscheinlichkeitsth. 31, 235–253.CrossRefGoogle Scholar
Kurtz, T. G. (1980) Representation of a Markov process as a multivariate time change. Ann. Prob. 8, 682–715.CrossRefGoogle Scholar
Meyer, P. A. (1971) Démonstration simplifée d'un théorème Knight. In Séminaire de Probabilities V, Springer Lecture Notes in Maths. 191, 191–195.Google Scholar
Papangelou, F. (1972) Integrability of expected increments of point processes and a related random change of scale. Trans. Amer. Math. Soc. 165, 483–506.CrossRefGoogle Scholar
Watanabe, S. (1964) On discontinuous additive functionals and Lévy measures of a Markov process. Jap. J. Math. 34, 31–70.CrossRefGoogle Scholar