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Mixing of cluster point processes

Published online by Cambridge University Press:  14 July 2016

G. M. Laslett
G. M. Laslett*
Affiliation:
CSIRO Division of Mathematics and Statistics, South Melbourne
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Abstract

This paper deals with the preservation of strong forms of mixing of point processes under the clustering operation. Strong, ϕ -, and *-mixing possess an increasing degree of uniformity of their asymptotic independence, and this turns out to be a significant factor in their preservation. In particular, it is indicated that ϕ-mixing may be maintained only under very stringent conditions (bounded clusters), whereas strong mixing is maintained under considerably milder conditions.

Keywords

POINT PROCESSCLUSTER PROCESSMIXING
Type
Research Papers
Information
Journal of Applied Probability , Volume 15 , Issue 4 , December 1978 , pp. 715 - 725
Copyright
Copyright © Applied Probability Trust 1978 

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References

Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Daley, D. J. (1976) Queueing output processes. Adv. Appl. Prob. 8, 395–415.Google Scholar
Daley, D. J. and Vere-Jones, D. (1972) A summary of the theory of point processes. In Stochastic Point Processes , ed. Lewis, P. A. W. Wiley, New York, 299–383.Google Scholar
Kemp, C. D. and Kemp, A. W. (1965) Some properties of the ‘Hermite’ distribution. Biometrika 52, 381–394.Google Scholar
Kerstan, J., Matthes, K. and Mecke, J. (1974) Unbegrenzt teilbare Punktprozesse. Akademie Verlag, Berlin. English translation (1977) Wiley, New York.Google Scholar
Laslett, G. M. (1975) Some Problems Concerning Cluster Processes and Other Point Processes. Ph.D. Thesis, The Australian National University, Canberra.Google Scholar
Milne, R. K. (1970) Identifiability for random translations of Poisson processes. Z. Wahrscheinlichkeitsth. 15, 195–201.Google Scholar
Oodaira, H. and Yoshihara, K. (1971) The law of the iterated logarithm for stationary processes satisfying mixing conditions. Kodai Math. Sem. Rep. 23, 311–334.Google Scholar
Philipp, W. (1969) The central limit theorem for mixing sequences of random variables. Z. Wahrscheinlichkeitsth. 12, 155–171.Google Scholar
Rosenblatt, M. (1956) A central limit theorem and strong mixing condition. Proc. Natn. Acad. Sci. USA 42, 43–47.Google Scholar
Westcott, M. (1971) On existence and mixing results for cluster point processes. J. R. Statist. Soc. B 33, 290–300.Google Scholar
Westcott, M. (1973) Results in the asymptotic and equilibrium theory of Poisson cluster processes. J. Appl. Prob. 10, 807–823.CrossRefGoogle Scholar