Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Estimation of Palm measures of stationary point processes
Download PDF
Download PDF
  • Published: March 1987

Estimation of Palm measures of stationary point processes

  • Alan F. Karr1 

Probability Theory and Related Fields volume 74, pages 55–69 (1987)Cite this article

  • 148 Accesses

  • 9 Citations

  • Metrics details

Summary

Estimators of the Palm measure of a stationary point process on a finite-dimensional Euclidean space are developed and shown to be strongly uniformly consistent. From them, similarly consistent estimators of reduced moment measures, the spectral measure, the spectral density function and the underlying probability measure itself are derived. Normal and Poisson approximations to distributions of estimators are presented. Application is made to the problem of combined inference and linear state estimation.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Bartlett, M.S.: The spectral analysis of point processes. J. Roy. Statist. Soc. B25, 264–296 (1963)

    Google Scholar 

  2. Bartlett, M.S.: The spectral analysis of two-dimensional point processes. Biometrika51, 299–311 (1964)

    Google Scholar 

  3. Bartlett, M.S.: The spectral analysis of line processes. Proc. Fifth Berkeley Sympos. Math. Statist. Probab.3, 135–153. University of California (1967)

    Google Scholar 

  4. Brillinger, D.R.: The spectral analysis of stationary interval functions. Proc. Sixth Berkeley Sympos. Math. Statist. Probab.1, 483–513. University of California (1972)

    Google Scholar 

  5. Brillinger, D.R.: Statistical inference for stationary point processes. In: Puri, M.L. (ed.) Stochastic processes and related topics. New York: Academic Press 1975

    Google Scholar 

  6. Çinlar, E.: Superposition of point processes. In: Lewis, P.A.W. (ed.) Stochastic point processes. New York: Wiley 1972

    Google Scholar 

  7. Cox, D.R., Lewis, P.A.W.: The statistical analysis of series of events. London: Chapman and Hall 1966

    Google Scholar 

  8. Daley, D.J.: Spectral properties of weakly stationary point processes. J. Roy. Statist. Soc. B33, 406–428 (1971)

    Google Scholar 

  9. Itô, K.: Stationary random distributions. Proc. Mem. Sci. Univ. Kyoto A28, 209–223 (1955)

    Google Scholar 

  10. Jolivet, E.: Central limit theorem and convergence of empirical processes for stationary point processes. In: Bartfai, P., Tomko, J. (eds.) Point processes and queueing problems. Amsterdam: North-Holland 1981

    Google Scholar 

  11. Kallenberg, O.: Random measures, 3rd edn. Berlin: Akademie-Verlag and New York: Academic Press 1983

    Google Scholar 

  12. Karr, A.F.: Classical limit theorems for measure-valued Markov processes. J. Multivariate Anal.9, 234–247 (1979)

    Google Scholar 

  13. Karr, A.F.: Inference for thinned point processes, with application to Cox processes. J. Multivariate Anal.16, 368–392 (1985)

    Google Scholar 

  14. Krickeberg, K.: Moments of point processes. In: Harding, E.F., Kendall, M.G. (eds.) Stochastic geometry. New York: Wiley 1974

    Google Scholar 

  15. Krickeberg, K.: Processus ponctuels en statistique. Lect. Notes Math.929, 205–313. Berlin Heidelberg New York: Springer 1982

    Google Scholar 

  16. Leonov, V.P., Shiryayev, A.N.: On a method of calculation of semi-invariants. Theor. Probab. Appl.4, 319–329 (1959)

    Google Scholar 

  17. Neveu, J.: Processus ponctuels. Lect. Notes Math.598, 249–447. Berlin Heidelberg New York: Springer 1977

    Google Scholar 

  18. Nguyen, X.X., Zessin, H.: Ergodic theorems for spatial processes. Z. Wahrscheinlichkeitstheor. Verw. Geb.48, 133–158 (1979)

    Google Scholar 

  19. Rudin, W.: Functional analysis. New York: McGraw-Hill 1973

    Google Scholar 

  20. Vere-Jones, D.: An elementary approach to the spectral theory of stationary random measures. In: Harding, E.F., Kendall, M.G. (eds.) Stochastic geometry. New York: Wiley 1974

    Google Scholar 

  21. Yoshida, K.: Functional analysis. Berlin Heidelberg New York: Springer 1968

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, The Johns Hopkins University, 21218, Baltimore, MD, USA

    Alan F. Karr

Authors
  1. Alan F. Karr
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Research supported by Air Force Office of Scientific Research, AFSC, grant 82-0029C. The United States Government is authorized to reproduce and distribute reprints for governmental purposes

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Karr, A.F. Estimation of Palm measures of stationary point processes. Probab. Th. Rel. Fields 74, 55–69 (1987). https://doi.org/10.1007/BF01845639

Download citation

  • Received: 26 March 1984

  • Revised: 07 April 1986

  • Issue Date: March 1987

  • DOI: https://doi.org/10.1007/BF01845639

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Density Function
  • Stochastic Process
  • Probability Measure
  • Probability Theory
  • Spectral Density
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature