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.
Article PDF
Similar content being viewed by others
References
Bartlett, M.S.: The spectral analysis of point processes. J. Roy. Statist. Soc. B25, 264–296 (1963)
Bartlett, M.S.: The spectral analysis of two-dimensional point processes. Biometrika51, 299–311 (1964)
Bartlett, M.S.: The spectral analysis of line processes. Proc. Fifth Berkeley Sympos. Math. Statist. Probab.3, 135–153. University of California (1967)
Brillinger, D.R.: The spectral analysis of stationary interval functions. Proc. Sixth Berkeley Sympos. Math. Statist. Probab.1, 483–513. University of California (1972)
Brillinger, D.R.: Statistical inference for stationary point processes. In: Puri, M.L. (ed.) Stochastic processes and related topics. New York: Academic Press 1975
Çinlar, E.: Superposition of point processes. In: Lewis, P.A.W. (ed.) Stochastic point processes. New York: Wiley 1972
Cox, D.R., Lewis, P.A.W.: The statistical analysis of series of events. London: Chapman and Hall 1966
Daley, D.J.: Spectral properties of weakly stationary point processes. J. Roy. Statist. Soc. B33, 406–428 (1971)
Itô, K.: Stationary random distributions. Proc. Mem. Sci. Univ. Kyoto A28, 209–223 (1955)
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
Kallenberg, O.: Random measures, 3rd edn. Berlin: Akademie-Verlag and New York: Academic Press 1983
Karr, A.F.: Classical limit theorems for measure-valued Markov processes. J. Multivariate Anal.9, 234–247 (1979)
Karr, A.F.: Inference for thinned point processes, with application to Cox processes. J. Multivariate Anal.16, 368–392 (1985)
Krickeberg, K.: Moments of point processes. In: Harding, E.F., Kendall, M.G. (eds.) Stochastic geometry. New York: Wiley 1974
Krickeberg, K.: Processus ponctuels en statistique. Lect. Notes Math.929, 205–313. Berlin Heidelberg New York: Springer 1982
Leonov, V.P., Shiryayev, A.N.: On a method of calculation of semi-invariants. Theor. Probab. Appl.4, 319–329 (1959)
Neveu, J.: Processus ponctuels. Lect. Notes Math.598, 249–447. Berlin Heidelberg New York: Springer 1977
Nguyen, X.X., Zessin, H.: Ergodic theorems for spatial processes. Z. Wahrscheinlichkeitstheor. Verw. Geb.48, 133–158 (1979)
Rudin, W.: Functional analysis. New York: McGraw-Hill 1973
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
Yoshida, K.: Functional analysis. Berlin Heidelberg New York: Springer 1968
Author information
Authors and Affiliations
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
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
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01845639