Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-25T07:41:25.409Z Has data issue: false hasContentIssue false
  • English
  • Français

The coincidence approach to stochastic point processes

Published online by Cambridge University Press:  01 July 2016

Odile Macchi
Odile Macchi*
Affiliation:
Laboratoire d'Etude des Phénomènes Aléatoires, Université de Paris-Sud
Get access Rights & Permissions [Opens in a new window]

Abstract

The structure of the probability space associated with a general point process, when regarded as a counting process, is reviewed using the coincidence formalism. The rest of the paper is devoted to the class of regular point processes for which all coincidence probabilities admit densities. It is shown that their distribution is completely specified by the system of coincidence densities. The specification formalism is stressed for ‘completely’ regular point processes. A construction theorem gives a characterization of the system of coincidence densities of such a process. It permits the study of most models of point processes. New results on the photon process, a particular type of conditioned Poisson process, are derived. New examples are exhibited, including the Gauss-Poisson process and the ‘fermion’ process that is suitable whenever the points are repulsive.

Keywords

STOCHASTIC POINT PROCESSJOINT COUNTING EVENTCOINCIDENCE PROBABILITYCOINCIDENCE DENSITYEXCLUSION PROBABILITY DENSITYREGULAR PROCESSPHOTON PROCESSGAUSS-POISSON PROCESSFERMION PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 7 , Issue 1 , March 1975 , pp. 83 - 122
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

[0] Macchi, O. (1972) Processus Ponctuels et Coincidences — Contribution à l'Etude Théorique des Processus Ponctuels — Applications à l'Optique Statistique et aux Communications Optiques. Thèse d'Etat. Orsay, France.Google Scholar
[1] Cox, D. R. (1962) Renewal Theory. Methuen, London.Google Scholar
[2] Cox, D. R. and Lewis, P. A. W. (1966) The Statistical Analysis of Series of Events. Methuen, London.CrossRefGoogle Scholar
[3] McFadden, J. (1962) On the lengths of intervals in a stationary point process. J. R. Statist. Soc. B 24, 364–382.Google Scholar
[4] Beutler, F. J. and Leneman, O. A. Z. (1966) The theory of stationary point processes. Acta Math. 116, 159–197.CrossRefGoogle Scholar
[5] Moyal, J. E. (1962) The general theory of stochastic population processes. Acta Math. 108, 1–31.Google Scholar
[6] Ryll-Nardzewski, C. (1961) Remarks on processes of calls. Proc. Fourth Berkeley Symp. Math. Statist. Prob. 2, 445–465 Google Scholar
[7] Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.Google Scholar
[8] Matthes, K. (1963) Stationäre zufälligepunktfolgen I. Jber. Deutsch Math. Verein 66, 66–79.Google Scholar
[9] Harris, T. E. (1968) Counting measures, monotone random set functions. Z. Wahrscheinlichkeitsth. 10, 102–119.Google Scholar
[10] Fortet, R. (1968) Sur les répartitions ponctuelles aléatoires, en particulier de Poisson. Ann. Inst. Henri Poincaré 4, p. 99.Google Scholar
[11] Kuznetsov, P. I. and Stratanovich, R. L. (1956) On the mathematical theory of correlated random points Izv. Akad. Nauk. SSSR. Ser. Mat. 20, 167–178. Translated into English in Vol. 7 of Selected Translations in Mathematical Statistics and Probability. Google Scholar
[12] Nawrotski, K. (1962) Eine Grenzwertsatz für homogene zufällige Punktfolgen. Math. Nachr. 24, 210–217.Google Scholar
[13] Daley, D. J. and Vere-Jones, D. (1972) A summary of the theory of point processes. Stochastic Point Processes: Statistical Analysis, Theory and Applications. Lewis, P. A. W., Editor. 299–383.Google Scholar
[14] Neveu, J. (1970) Bases Mathématiques du Calcul des Probabilités. 2 édition, Masson, Paris.Google Scholar
[15] Boas, R. P. (1954) Entire Functions. Academic Press, New York.Google Scholar
[16] Cox, D. R. (1955) Some statistical methods connected with series of events. J. R. Statist. Soc. B. 17, 129–164.Google Scholar
[17] Bartlett, M. S. (1963) The spectral analysis of point processes. J. R. Statist. Soc. B 25, 264–296.Google Scholar
[18] Gaver, D. P. (1963) Random hazard in reliability problems. Technometrics 5, 211–226 CrossRefGoogle Scholar
[19] Mandel, L. and Wolf, E. (1965) Coherence properties of optical fields. Rev. Mod. Phys. 37, 231–287.Google Scholar
[20] Rudemo, M. (1972) Doubly-stochastic Poisson processes and process control. Adv. Appl. Prob. 4, 318–338.CrossRefGoogle Scholar
[21] Cramér, H. (1968) On streams of random events. Symposium on Risk Theory, Stockholm.Google Scholar
[22] Snyder, D. L. (1972) Filtering and detection for doubly-stochastic Poisson processes. I.E.E.E. Trans. Inform. Theo. Vol. IT-18, 91–102.Google Scholar
[23] Grandell, J. (1971) On stochastic processes generated by a stochastic intensity function. Skand. Aktuartidskr. 54, 204–240.Google Scholar
[24] Helstrom, C. W. (1964) The distribution of photoelectric counts from partially polarized Gaussian light. Proc. Phys. Soc. 83, 777–782.CrossRefGoogle Scholar
[25] Macchi, O. (1971) Distribution statistique des instants d'émission des photoélectrons d'une lumière thermique. C.R.A.S. série A 272, 437–440.Google Scholar
[26] Picinbono, B., Bendjaballah, C. and Pouget, J. (1970) Photoelectron shot noise. J. Math. Phys. 11, 2166–2176.CrossRefGoogle Scholar
[27] Newman, D. S. (1970) A new family of point processes which are characterized by their second moment properties. J. Appl. Prob. 7, 338–358.CrossRefGoogle Scholar
[28] Milne, R. K. and Westcott, M. (1972) Further results for Gauss-Poisson processes Adv. Appl. Prob. 4, 151–176.Google Scholar
[29] Loève, M. (1963) Probability Theory. Van Nostrand, 3rd edition.Google Scholar
[30] Van Trees, H. L. (1968) Detection, Estimation and Modulation Theory, Part. I. Wiley, New York.Google Scholar
[31] Benard, C. (1970) Fluctuations of beams of quantum particles. Phys. Rev. A-2, No. 5, p. 2140.Google Scholar
[32] Siegart, A. F. J. (1957) Systematic approach to a class of problems in the theory of noise and other random phenomena. Part II, IRE Trans. on Infor. Theo. IT-3. 38–43.CrossRefGoogle Scholar
[33] Macchi, O. (1972) Estimation and detection of weak optical signals. I.E.E.E. Trans. IT-18 no. 5, 562–573.Google Scholar
[34] Benard, C. and Macchi, O. (1973) Detection and emission processes of quantum particles in a chaotic state. J. Math, Phys. 14, 155–167.Google Scholar
[35] Beckenbach, E. F. and Bellman, R. (1965) Inequalities. 2nd edition, Springer-Verlag, New York.CrossRefGoogle Scholar
[36] Courant, R. and Hilbert, D. (1953) Methods of Mathematical Physics, Vol. I. Interscienence, New York.Google Scholar
[37] Macchi, O. (1972) Etude d'un processus ponctuel caractérisé par ses multicoincidences. C.R.A.S. série A 271, 660–663.Google Scholar
[38] Macchi, O. (1971) Stochastic processes and multicoincidences. I.E.E.E. Trans IT-17 no. 1, 2–7.Google Scholar
[39] Riesz, F. and Nagy, B. SZ. (1968) Leçons d'Analyse Fonctionnelle. 5th Edition. Gauthier-Villars, Paris.Google Scholar