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A characterization of the Poisson process using forward recurrence times

Published online by Cambridge University Press:  24 October 2008

Valerie Isham ,
D. N. Shanbhag  and
M. Westcott
Valerie Isham
Affiliation:
Department of Mathematics, Imperial College, London
D. N. Shanbhag
Affiliation:
Department of Probability and Statistics, University of Sheffield
M. Westcott
Affiliation:
Department of Mathematics, Imperial College, London
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Extract

Consider a renewal process on the nonnegative real line with non-arithmetic distribution function F(x). Denote by V(x; t) the distribution function of the forward recurrence time from t, t ≤ 0. If t is chosen at random with distribution function Ф(t), the corresponding unconditional forward recurrence time has distribution function

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 78 , Issue 3 , November 1975 , pp. 513 - 516
Copyright
Copyright © Cambridge Philosophical Society 1975

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References

REFERENCES

(1)Blackwell, D.A renewal theorem. Duke Math. J. 15 (1948), 145–150.CrossRefGoogle Scholar
(2)Chung, K. L.The Poisson process as a renewal process. Periodica Math. Hung. 2 (1972), 41–48.CrossRefGoogle Scholar
(3)Çinlar, E. and Jagers, P.Two mean values which characterise the Poisson process. J. Appl. Probability 10 (1973), 678–681.CrossRefGoogle Scholar
(4)Feller, W.An Introduction to Probability Theory and its Applications, vol. II (2nd ed.), Wiley (1971).Google Scholar
(5)Kotz, S. and Johnson, N. L.A characterization of the exponential distribution by a waiting time property. Communications in Statistics 3 (1974), 257–258.Google Scholar
(6)Slivnyak, I. M.Some properties of stationary flows of homogeneous random events. Theor. Probability Appl. 7 (1962), 336–341.CrossRefGoogle Scholar
(7)Smith, W. L.On renewal theory, counter problems and quasi-Poisson processes. Proc. Cambridge Philos. Soc. 53 (1957), 175–193.CrossRefGoogle Scholar