Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-19T01:59:25.519Z Has data issue: false hasContentIssue false
  • English
  • Français

Dependent thinning of point processes

Published online by Cambridge University Press:  14 July 2016

Valerie Isham
Valerie Isham*
Affiliation:
University College London
*
∗Postal address: Department of Statistical Science, University College London, Gower St., London WCIE 6BT, U.K.
Get access Rights & Permissions [Opens in a new window]

Abstract

A point process, N, on the real line, is thinned using a k -dependent Markov sequence of binary variables, and is rescaled. Second-order properties of the thinned process are described when k = 1. For general k, convergence to a compound Poisson process is demonstrated.

Keywords

THINNED POINT PROCESSCOMPOUND POISSON PROCESS
Type
Research Papers
Information
Journal of Applied Probability , Volume 17 , Issue 4 , December 1980 , pp. 987 - 995
Copyright
Copyright © Applied Probability Trust 

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

Belyaev, Yu. K. (1963) Limit theorems for dissipative flows. Theory Prob. Appl. 8, 165–173.Google Scholar
Goldman, J. R. (1967) Stochastic point processes: limit theorems. Ann. Math. Statist. 38, 771–779.Google Scholar
Kallenberg, O. (1975) Limits of compound and thinned point processes, J. Appl. Prob. 12, 269–278.Google Scholar
Nawrotski, K. (1962) Ein Grenzwertsatz für homogene züfallige Punktfolgen (Verall-gemeinerung eines Satzes von A. Rényi). Math. Nachr. 24, 201–217.Google Scholar
Papangelou, F. (1972) Integrability of expected increments of point processes and a related random change of scale. Trans. Amer. Math. Soc. 165, 483–506.Google Scholar
Renyi, A. (1956) A characterisation of Poisson processes (in Hungarian). Magyar Tud. Akad. Mat. Kutató Int. Közl. 1, 519–527. (Translated in Selected papers of Alfréd Rényi Vol. 1, ed. Turán, Pál, Akademiai Kiadó, Budapest (1976), 622–628.Google Scholar
Rudemo, M. (1973) On a random transformation of a point process to a Poisson process. In Mathematics and Statistics: Essays in honour of Harald Bergström, ed. Jagers, P. and Råde, L., Göteborg, pp. 79–85.Google Scholar
Westcott, M. (1976) Simple proof of a result on thinned point processes. Ann. Prob. 4, 89–90.Google Scholar