Skip to main content
×
×
Home

The second-order analysis of stationary point processes

  • B. D. Ripley (a1)
Abstract

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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      The second-order analysis of stationary point processes
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      The second-order analysis of stationary point processes
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      The second-order analysis of stationary point processes
      Available formats
      ×
Copyright
References
Hide All
Bartlett, M. S. (1964) The spectral analysis of two-dimensional point processes. Biometrika 51, 299311.
Bartlett, M. S. (1967) The spectral analysis of line processes. Proc. 5th Berkeley Symp. Math. Statist. Prob. 3, 135153.
Bartlett, M. S. (1974) The statistical analysis of spatial pattern. Adv. Appl. Prob. 6, 336358.
Bhabha, H. J. (1950) On the stochastic theory of continuous parameter systems and its application to electron cascades. Proc. R. Soc. Lond. A 202, 301322.
Bourbaki, N. (1963) Intégration VII. Hermann, Paris.
Bourbaki, N. (1966) General Topology. Hermann, Paris.
Chriŝtensen, J. P. R. (1974) Topology and Borei Structure. North Holland, Amsterdam.
Cox, D. R. and Lewis, P. A. W. (1966) The Statistical Analysis of Series of Events. Methuen, London.
Davidson, R. (1970) Construction of line processes: second order properties. In Stochastic Geometry , ed. Harding, E. F. and Kendall, D. G., Wiley, London, 5575.
Gel'fand, I. M. and Vilenkin, N. Ya. (1964) Generalized Functions , Vol. 4. Academic Press, New York.
Glass, L. and Tobler, W. R. (1971) Uniform distribution of objects in a homogeneous field, cities on a plain. Nature 233, No. 5314, 6768.
Julesz, B. (1975) Experiments in the visual perception of data. Scientific American , April, 3443.
Kendall, D. G. (1974) An introduction to stochastic geometry. In Harding, and Kendall, (1974), 39.
Krickeberg, K. (1970) Invariance properties of the correlation measure of line processes. In Harding, and Kendall, (1974), 7688.
Krickeberg, K. (1973) Moments of point processes. In Harding, and Kendall, (1974), 89113.
Matern, B. (1960) Spatial variation. Meddelanden från Statens Skogsforskningsinstitut 49:5.
Mecke, J. (1967) Stationäre zufällige Maße auf lokalkompacten Abelschen Gruppen. Z. Wahrscheinlichkeitsth. 9, 3658.
Moyal, J. E. (1962) The general theory of stochastic population processes. Acta. Math. 108, 131.
Nachbin, L. (1967) The Haar Integral. Van Nostrand, Princeton.
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, 419440.
Ramakrishnan, A. (1950) Stochastic processes relating to particles distributed in a continuous infinity of states. Proc. Camb. Phil. Soc. 46, 595602.
Ripley, B. D. (1976a) The disintegration of invariant measures. Math. Proc. Camb. Phil. Soc. , To appear.
Ripley, B. D. (1976b) Locally finite random sets. Ann. Prob. Submitted for publication.
Ripley, B. D. (1976C) The foundations of stochastic geometry. Ann. Prob. Submitted for publication.
Santaló, L. A. (1953) Introduction to Integral Geometry. Hermann, Paris.
Serra, J. (1972) Stereology and structuring elements. J. Microscopy 95, 93103.
Silverman, B. W. (1976) Limit theorems for dissociated random variables. J. Appl. Prob. Submitted for publication.
Strauss, D. J. (1975) A model for clustering. Biometrika 62, 467475.
Vere-Jones, D. (1968) Some applications of probability generating functionals to the study of input-output streams. J. R. Statist. Soc. B 30, 321333.
Vere-Jones, D. (1970) Stochastic models for earthquake occurrence. J. R. Statist. Soc. B. 32, 162.
Vere-Jones, D. (1971) The covariance measure of a weakly stationary random measure. J. R. Statist. Soc. B 33, 426428.
Vere-Jones, D. (1974) An elementary approach to the spectral theory of stationary point processes. In Harding, and Kendall, (1974), 307321.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed