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The probability generating functional

Published online by Cambridge University Press:  09 April 2009

M. Westcott
M. Westcott
Affiliation:
Department of StatisticsAustralian National UniversityCanberra
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This paper is concerned with certain aspects of the theory and application of the probability generating functional (p.g.fl) of a point process on the real line. Interest in point processes has increased rapidly during the last decade and a number of different approaches to the subject have been expounded (see for example [6], [11], [15], [17], [20], [25], [27], [28]). It is hoped that the present development using the p.g.ff will calrify and unite some of these viewpoints and provide a useful tool for solution of particular problems.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 14 , Issue 4 , December 1972 , pp. 448 - 466
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Bartlett, M. S., ‘The spectral analysis of point processes’, J. Roy. Statist. Soc. Ser. B. 25 (1963), 264–296.Google Scholar
[2]Bartlett, M. S., Stochastic Processes (C. U. P., 2nd ed. 1966).Google Scholar
[3]Cox, D. R. and Lewis, P. A. W., The Statistical Analysis of Series of Events (Methuen, 1966).CrossRefGoogle Scholar
[4]Grigelionis, B., ‘On the convergence of random step processes to a Poisson process’, Th. Probab. Appl. 8 (1963), 177–182.CrossRefGoogle Scholar
[5]Halmos, P. R., Measure Theory (Van Nostrand, 1950).CrossRefGoogle Scholar
[6]Harris, T. E., The Theory of Branching Processes (Springer, 1963).Google Scholar
[7]Jacobs, K., ‘On Poincaré's recurrence theorem’, Proc. 5th Berkeley Symp. 2, II (1965), 375–404.Google Scholar
[8]Jiřina, M., ‘Asymptotic behaviour of measure-valued branching processes’, Rozpravy Českosl. Akad. Věd. R. Mat. Privodnich Věd. 76, 3 (1966).Google Scholar
[9]Kerstan, J. and Matthes, K., ‘Stationä rezufällige Punktfolgen II’, J-ber. Deutsch. Math. Verein. 66 (1963), 106–118.Google Scholar
[10]Kerstan, J. and Matthes, K., ‘Ergodische unbegrenzt teilbare stationäre zufällige Punktfolgen’, Trans. 4th Prague Conf. (1965), 399–415.Google Scholar
[11]Khintchine, A. Ya., Mathematical Methods in the Theory of Queueing (Griffin, 2nd ed., 1969).Google Scholar
[12]Lee, P. M., ‘Infinitely divisible stochastic processes’, Z. Wahrscheinlichkeitstheorie verw. Geb. 7 (1967), 147–160.CrossRefGoogle Scholar
[13]Lee, P. M., ‘Some examples of infinitely divisible point processes’, Stud. Sci. Math. Hung. 3 (1968), 219–224.Google Scholar
[14]Leonov, V. P., ‘Applications of the characteristic functional and semi-invariants to the ergodic theory of stationary processes’, Dokl. Acad. Nauk SSSR (1960), 523–526.Google Scholar
[15]Matthes, K., ‘Stationäre zufällige Puntkfolgen I’, J-ber. Deutsch Math. Verein. 66 (1963), 66–79.Google Scholar
[16]Matthes, K., ‘Unbeschränkt teilbare Verteilungsgesetze stationärer zufälliger Punktfolgen’, Wiss. Z. Hochschule Elektro Ilmenau 9 (1963), 225–238.Google Scholar
[17]Mecke, J., ‘Stationäre zufällige Masse auf Lokalkompakten Abelschen Gruppen’, Z. Wahrscheinlichkeitstheorie verw. Geb. 9 (1967), 36–58.CrossRefGoogle Scholar
[18]Milne, R. K., ‘Simple proofs of some theorems on point processes’, Ann. Math. Statist. 42 (1971) 368 372.CrossRefGoogle Scholar
[19]Moran, P. A. P., An Introduction to Probability Theory (O. U. P., 1968).Google Scholar
[20]Moyal, J. E., ‘The general theory of stochastic population processes’, Acta Math. 108 (1962), 1–31.CrossRefGoogle Scholar
[21]Moyal, J. E., ‘Multiplicative population chains’, Proc. Roy. Soc. A 266 (1962), 518–526.Google Scholar
[22]Nawrotzki, K., ‘Eine Grenzwertsatz für homogene zufällige Punktfolgen’, Math. Nachr. 24 (1962), 201–217.CrossRefGoogle Scholar
[23]Rosenblatt, M., Random Processes (O. U. P. 1962).Google Scholar
[24]Ryll-Nardzewski, C., ‘Remarks on the Poisson stochastic process (III)’, Studia. Math. 14 (1954), 314–318.CrossRefGoogle Scholar
[25]Ryll-Nardzewski, C., ‘Remarks on processes of calls’, Proc. 4th Berkeley Symp. 2 (1960), 455–465.Google Scholar
[26]Shiryaev, A. N., ‘Some problems in the spectral theory of higher order moments I’, Th. Probab. Appl. 5 (1960), 265–284.CrossRefGoogle Scholar
[27]Slivnyak, I. M., ‘Some properties of stationary flows of homogeneous random events’, Th. Probab. Appl. 7 (1962), 336–341 (see also 9, (1964), 168).CrossRefGoogle Scholar
[28]Vere-Jones, D., ‘Some applications of probability generating functionals to the study of input-output streams’, J. Roy. Statist. Soc. Ser. B 30 (1968), 321–333.Google Scholar
[29]Vere-Jones, D., ‘Stochastic models for earthquake occurrence’, J. Roy. Statist. Soc. Ser. B 32 (1970), 1–62.Google Scholar
[30]Westcott, M., ‘Identifiability in linear processes’, Z. Wahrscheinlichkeistheorie verw. Geb. 16 (1970), 39–46.CrossRefGoogle Scholar
[31]Wescott, M.,. ‘On existence and mixing results for cluster point processe’, J. Roy. Statist. Soc. Ser. B. 33 (1971), 290–300.Google Scholar
[32]Daley, D. J. and Vere-Jones, D., ‘A summary of the theory of point processes’, in Stochastic Point Processes: Statistical Analysis, Theory and Application, edited by Lewis, P. A. W.299–383 (Wley, 1972).Google Scholar