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On doubly stochastic Poisson processes

Published online by Cambridge University Press:  24 October 2008

J. F. C. Kingman
J. F. C. Kingman
Affiliation:
Pembroke College, Cambridge
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Extract

The class of stationary point processes known as ‘doubly stochastic Poisson processes’ was introduced by Cox (2) and has been studied in detail by Bartlett (1). It is not clear just how large this class is, and indeed it seems to be a problem of some difficulty to decide of a general stationary point process whether or not it can be represented as a doubly stochastic Poisson process. (A few simple necessary conditions are known. For instance, Cox pointed out in the discussion to (1) that a double stochastic Poisson process must show more ‘dispersion’ than the Poisson process. Such conditions are very far from being sufficient.) The main result of the present paper is a solution of the problem for the special case of a renewal process, justifying an assertion made in the discussion to (1).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 60 , Issue 4 , October 1964 , pp. 923 - 930
Copyright
Copyright © Cambridge Philosophical Society 1964

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References

REFERENCES

(1)Bartlett, M. S.The spectral analysis of point processes. J. Roy. Statist. Soc. Ser. B, 25 (1963), 264–296.Google Scholar
(2)Cox, D. R.Some statistical models connected with series of events. J. Roy. Statist. Soc. Ser. B, 17 (1955), 129–164.Google Scholar
(3)Cox, D. R.Renewal theory (Methuen; London, 1962).Google Scholar
(4)Doob, J. L.Stochastic processes (Wiley; New York, 1953).Google Scholar
(5)Kendall, D. G.Extreme-point methods in stochastic analysis. Z. Wahrscheinlichkeits- theorie und Verw. Gebiete, 1 (1963), 295–300.CrossRefGoogle Scholar
(6)Kingman, J. F. C.A continuous time analogue of the theory of recurrent events. Bull. American Math. Soc. 69 (1963), 268–272.Google Scholar
(7)Kingman, J. F. C.Poisson counts for random sequences of events. Ann. Math. Statist. 34 (1963), 1217–1232.CrossRefGoogle Scholar
(8)Kingman, J. F. C.The stochastic theory of regenerative events. Z. Wahrscheinlichkeits- theorie und Verw. Gebiete, 2 (1964), 180–224.CrossRefGoogle Scholar
(9)Kingman, J. F. C.Linked systems of regenerative events. Proc. London Math. Soc. (to appear).Google Scholar
(10)Ryll-Nardzewski, C. Remarks on processes of calls, from Proc. 4th Berkeley Sympos. Math. Statist, and Prob., vol. ii (Berkeley, 1961).Google Scholar