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The second-order analysis of stationary point processes

Published online by Cambridge University Press:  14 July 2016

B. D. Ripley
B. D. Ripley*
Affiliation:
University of Cambridge
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Abstract

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This paper provides a rigorous foundation for the second-order analysis of stationary point processes on general spaces. It illuminates the results of Bartlett on spatial point processes, and covers the point processes of stochastic geometry, including the line and hyperplane processes of Davidson and Krickeberg. The main tool is the decomposition of moment measures pioneered by Krickeberg and Vere-Jones. Finally some practical aspects of the analysis of point processes are discussed.

Keywords

MOMENT MEASURESSTATIONARY POINT PROCESSESSPATIAL POINT PROCESSESLINE PROCESSESHYPERPLANE PROCESSESSTOCHASTIC GEOMETRY
Type
Research Papers
Information
Journal of Applied Probability , Volume 13 , Issue 2 , June 1976 , pp. 255 - 266
Copyright
Copyright © Applied Probability Trust 1976 

References

Bartlett, M. S. (1964) The spectral analysis of two-dimensional point processes. Biometrika 51, 299–311.CrossRefGoogle Scholar
Bartlett, M. S. (1967) The spectral analysis of line processes. Proc. 5th Berkeley Symp. Math. Statist. Prob. 3, 135–153.Google Scholar
Bartlett, M. S. (1974) The statistical analysis of spatial pattern. Adv. Appl. Prob. 6, 336–358.CrossRefGoogle Scholar
Bhabha, H. J. (1950) On the stochastic theory of continuous parameter systems and its application to electron cascades. Proc. R. Soc. Lond. A 202, 301–322.Google Scholar
Bourbaki, N. (1963) Intégration VII. Hermann, Paris.Google Scholar
Bourbaki, N. (1966) General Topology. Hermann, Paris.Google Scholar
Chriŝtensen, J. P. R. (1974) Topology and Borei Structure. North Holland, Amsterdam.Google Scholar
Cox, D. R. and Lewis, P. A. W. (1966) The Statistical Analysis of Series of Events. Methuen, London.CrossRefGoogle Scholar
Davidson, R. (1970) Construction of line processes: second order properties. In Stochastic Geometry , ed. Harding, E. F. and Kendall, D. G., Wiley, London, 55–75.Google Scholar
Gel'fand, I. M. and Vilenkin, N. Ya. (1964) Generalized Functions , Vol. 4. Academic Press, New York.Google Scholar
Glass, L. and Tobler, W. R. (1971) Uniform distribution of objects in a homogeneous field, cities on a plain. Nature 233, No. 5314, 67–68.CrossRefGoogle Scholar
Julesz, B. (1975) Experiments in the visual perception of data. Scientific American , April, 34–43.CrossRefGoogle Scholar
Kendall, D. G. (1974) An introduction to stochastic geometry. In Harding, and Kendall, (1974), 3–9.Google Scholar
Krickeberg, K. (1970) Invariance properties of the correlation measure of line processes. In Harding, and Kendall, (1974), 76–88.Google Scholar
Krickeberg, K. (1973) Moments of point processes. In Harding, and Kendall, (1974), 89–113.CrossRefGoogle Scholar
Matern, B. (1960) Spatial variation. Meddelanden från Statens Skogsforskningsinstitut 49:5.Google Scholar
Mecke, J. (1967) Stationäre zufällige Maße auf lokalkompacten Abelschen Gruppen. Z. Wahrscheinlichkeitsth. 9, 36–58.CrossRefGoogle Scholar
Moyal, J. E. (1962) The general theory of stochastic population processes. Acta. Math. 108, 1–31.CrossRefGoogle Scholar
Nachbin, L. (1967) The Haar Integral. Van Nostrand, Princeton.Google Scholar
Perkel, D. H., Gerstein, G. L. and Moore, G. P. (1967) Neuronal spike trains and stochastic point processes II. Simultaneous spike trains. Biophys. J. 7, 419–440.CrossRefGoogle ScholarPubMed
Ramakrishnan, A. (1950) Stochastic processes relating to particles distributed in a continuous infinity of states. Proc. Camb. Phil. Soc. 46, 595–602.CrossRefGoogle Scholar
Ripley, B. D. (1976a) The disintegration of invariant measures. Math. Proc. Camb. Phil. Soc. , To appear.CrossRefGoogle Scholar
Ripley, B. D. (1976b) Locally finite random sets. Ann. Prob. Submitted for publication.Google Scholar
Ripley, B. D. (1976C) The foundations of stochastic geometry. Ann. Prob. Submitted for publication.CrossRefGoogle Scholar
Santaló, L. A. (1953) Introduction to Integral Geometry. Hermann, Paris.Google Scholar
Serra, J. (1972) Stereology and structuring elements. J. Microscopy 95, 93–103.CrossRefGoogle Scholar
Silverman, B. W. (1976) Limit theorems for dissociated random variables. J. Appl. Prob. Submitted for publication.CrossRefGoogle Scholar
Strauss, D. J. (1975) A model for clustering. Biometrika 62, 467–475.CrossRefGoogle Scholar
Vere-Jones, D. (1968) Some applications of probability generating functionals to the study of input-output streams. J. R. Statist. Soc. B 30, 321–333.Google Scholar
Vere-Jones, D. (1970) Stochastic models for earthquake occurrence. J. R. Statist. Soc. B. 32, 1–62.Google Scholar
Vere-Jones, D. (1971) The covariance measure of a weakly stationary random measure. J. R. Statist. Soc. B 33, 426–428.Google Scholar
Vere-Jones, D. (1974) An elementary approach to the spectral theory of stationary point processes. In Harding, and Kendall, (1974), 307–321.Google Scholar