Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-12-01T00:58:22.785Z Has data issue: false hasContentIssue false
  • English
  • Français

Limits of compound and thinned point processes

Published online by Cambridge University Press:  14 July 2016

Olav Kallenberg
Olav Kallenberg*
Affiliation:
University of Göteborg
Get access Rights & Permissions [Opens in a new window]

Abstract

Let η =Σjδτj be a point process on some space S and let β,β1,β2, … be identically distributed non-negative random variables which are mutually independent and independent of η. We can then form the compound point process ξ = Σjβjδτj which is a random measure on S. The purpose of this paper is to study the limiting behaviour of ξ as . In the particular case when β takes the values 1 and 0 with probabilities p and 1 –p respectively, ξ becomes a p-thinning of η and our theorems contain some classical results by Rényi and others on the thinnings of a fixed process, as well as a characterization by Mecke of the class of subordinated Poisson processes.

Keywords

COMPOUND AND THINNED POINT PROCESSESINFINITELY DIVISIBLE RANDOM MEASURESSUBORDINATED POISSON PROCESSESCONVERGENCE IN DISTRIBUTIONREGULARITY AND DIFFUSENESS
Type
Research Papers
Information
Journal of Applied Probability , Volume 12 , Issue 2 , June 1975 , pp. 269 - 278
Copyright
Copyright © Applied Probability Trust 1975 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Belyaev, Yu. K. (1963) Limit theorems for dissipative flows. Theor. Probad. Appl. 8, 165–173.CrossRefGoogle Scholar
[2] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
[3] Goldman, J. R. (1967) Stochastic point processes: limit theorems. Ann. Math. Statist. 38, 771–779.CrossRefGoogle Scholar
[4] Kallenberg, O. (1973) Characterization and convergence of random measures and point processes. Z. Wahrscheinlichkeitsth. 27, 9–21.CrossRefGoogle Scholar
[5] Kallenberg, O. (1973) Canonical representations and convergence criteria for processes with interchangeable increments. Z. Wahrscheinlichkeitsth. 27, 23–36.CrossRefGoogle Scholar
[6] Kerstan, J., Matthes, K. and Mecke, J. (1974) Unbegrenzt teilbare Punktprozesse. Akademie-Verlag, Berlin.Google Scholar
[7] Loève, M. (1963) Probability Theory , 3rd. ed. Van Nostrand, Princeton.Google Scholar
[8] Mecke, J. (1968) Eine characteristische Eigenschaft der doppelt stochastischen Poissonschen Prozesse. Z. Wahrscheinlichkeitsth. 11, 74–81.CrossRefGoogle Scholar
[9] Nawrotzki, K. (1962) Ein Grenzwertsatz für homogene zufällige Punktfolgen (Verallgemeinerung eines Satzes von A. Rényi). Math. Nachr. 24, 201–217.Google Scholar
[10] Rényi, A. (1956) A characterization of Poisson processes. (In Hungarian with summaries in Russian and English.) Magyar Tud. Akad. Mat. Kutató Int. Közl. 1, 519–527.Google Scholar
[11] Waldenfels, W.V. (1968) Characteristische Funktionale zufälliger Masse. Z. Wahrscheinlichkeitsth. 10, 279–283.CrossRefGoogle Scholar
[12] Kallenberg, O. (1975) Random Measures , Schriftenreihe des Zentralinstituts für Mathematik und Mechanic der AdW der DDR. Academic-Verlag, Berlin.Google Scholar