Hostname: page-component-68945f75b7-tmfhh Total loading time: 0 Render date: 2024-08-06T05:12:17.514Z Has data issue: false hasContentIssue false
  • English
  • Français

Further results for Gauss-Poisson processes

Published online by Cambridge University Press:  01 July 2016

R. K. Milne  and
M. Westcott
R. K. Milne*
Affiliation:
The Australian National University
M. Westcott*
Affiliation:
The Australian National University
*
∗ Now at London School of Economics.
∗∗ Now at Imperial College, University of London.
Get access Rights & Permissions [Opens in a new window]

Abstract

Newman (1970) introduced an interesting new class of point processes which he called Gauss-Poisson. They are characterized, in the most general case, by two measures. We determine necessary and sufficient conditions on these measures for the resulting point process to be well defined, and proceed to a systematic study of its properties. These include stationarity, ergodicity, and infinite divisibility. We mention connections with other classes of point processes and some statistical results. Our basic approach is through the probability generating functional of the process.

Keywords

POINT PROCESSESGAUSS-POISSON PROCESSESPROBABILITY GENERATING FUNCTIONALINFINITELY DIVISIBLE POINT PROCESSBIVARIATE POINT PROCESSDOUBLY STOCHASTIC POINT PROCESSPOISSON CLUSTERING PROCESSMIXING POINT PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 4 , Issue 1 , April 1972 , pp. 151 - 176
Copyright
Copyright © Applied Probability Trust 1972 

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

Bartlett, M. S. (1963) The spectral analysis of point processes. J. R. Statist. Soc. B 25, 264–296.Google Scholar
Belyaev, Yu. K. (1963) Limit theorems for dissipative flows. Theor. Probability Appl. 8, 165–173.CrossRefGoogle Scholar
Bochner, S. (1955) Harmonic Analysis and the Theory of Probability. Univ. of Calif. Press, Berkeley and Los Angeles.CrossRefGoogle Scholar
Cox, D. R. and Lewis, P. A W. (1966) The Statistical Analysis of Series of Events. Methuen, London.Google Scholar
Cox, D. R. and Lewis, P. A. W. (1970) Multivariate point processes. Proc. Sixth Berkeley Symp. Math. Statist. and Prob. (to appear).Google Scholar
Daley, D. J. (1971) Weakly stationary point processes and random measures. J. R. Statist. Soc. B 33 (to appear).Google Scholar
Feller, W. (1968) An Introduction to Probability Theory and its Applications. Vol. I 3rd ed. John Wiley and Sons Inc., New York.Google Scholar
Goldman, J. R. (1967a) Infinitely divisible point process in R n . J. Math. Anal. Appl. 17, 133–146.Google Scholar
Goldman, J. R. (1967b) Stochastic point processes; Limit theorems. Ann. Math. Statist. 38, 771–779.CrossRefGoogle Scholar
Halmos, P. R. (1950) Measure Theory. Van Nostrand, New York.Google Scholar
Hardy, G., Littlewood, J. E. and Pólya, G. (1934) Inequalities. Cambridge University Press, Cambridge.Google Scholar
Harris, T. E. (1963) The Theory of Branching Processes. Springer, Berlin.CrossRefGoogle Scholar
Kemp, A. W. and Kemp, C. D. (1965) Some properties of the ‘Hermlte’ distribution. Biometrika 52, 381–394.Google ScholarPubMed
Kemp, A. W. and Kemp, C. D. (1966) An alternative derivation of the Hermite distribution. Biometrika 53, 627–628.Google Scholar
Kerstan, J. and Matthes, K. (1964) Stationäre zufällige Punktfolgen II. J-ber. Deutsch. Math. Verein. 66, 106–118.Google Scholar
Kerstan, J. and Matthes, K. (1965) Ergodische unbegrenzt teilbare stationäre zufällige Punktfolgen. Trans. 4th Prague Conf. on Information Theory, Statistical Decision Functions, Random Processes. 399–415.Google Scholar
Khinchin, A. Ya. (1956) Sequences of chance events without after-effect. Theor. Probability Appl. 1, 1–15.CrossRefGoogle Scholar
Khinchin, A. Ya. (1969) Mathematical Methods in the Theory of Queueing. 2nd ed. Griffin, London.Google Scholar
Kingman, J. F. C. (1964) On doubly stochastic Poisson processes. Proc. Camb. Phil. Soc. 60, 923–930.Google Scholar
Kingman, J. F. C. (1967) Completely random measures. Pacific J. Math. 21, 59–78.Google Scholar
Kuznecov, P. I. and Stratonovich, R. L. (1956) On the mathematical theory of correlated random points. Izv. Akad. Nauk. SSSR Ser. Mat. 20, 167–178. Translated in Selected Translations in Mathematical Statistics and Probability Vol. 7, American Mathematical Society, 1968.Google Scholar
Lee, P. M. (1967) Infinitely divisible stochastic processes. Z. Wahrscheinlichkeitsth. 7, 147–160.CrossRefGoogle Scholar
Lévy, P. (1937) Sur les exponentielles de polynômes. Ann. Sci. École Norm. Sup. 54, 231–292.Google Scholar
Lewis, P. A. W. (1964) A branching Poisson process model for the analysis of computer failure patterns. J. R. Statist. Soc. B 26, 398–456.Google Scholar
Lewis, P. A. W. (1967) Non-homogeneous branching Poisson processes. J. R. Statist. Soc. B 29, 343–354.Google Scholar
Lewis, P. A. W. (1969) Asymptotic properties and equilibrium conditions for branching Poisson processes. J. Appl. Prob. 6, 355–371.CrossRefGoogle Scholar
Loève, M. (1960) Probability Theory. 2nd ed. Van Nostrand, New York.Google Scholar
Lukacs, E. (1960) Characteristic Functions. Griffin, London.Google Scholar
Matthes, K. (1963) Unbeschränkt teilbare Verteilungsgesetze stationärer zufälliger Punktfolgen. Wiss. Z. Hochsch. Electro. Ilmenau 9, 235–238.Google Scholar
Milne, R. K. (1970) Identifiability for random translations of Poisson processes. Z. Wahrscheinlichkeitsth. 15, 195–201.Google Scholar
Milne, R. K. (1971) Stochastic Analysis of Multivariate Point Processes. Ph. D. Thesis, The Australian National University, Canberra.Google Scholar
Moyal, J. E. (1962) The general theory of stochastic population processes. Acta Math. 108, 1–31.Google Scholar
Newman, D. S. (1970) A new family of point processes which are characterized by their second moment properties. J. Appl. Prob. 7, 338–358.Google Scholar
Prohorov, Yu. V. and Rozanov, Yu. A. (1969) Probability Theory. Springer, Berlin.Google Scholar
Ruben, H. (1964) Generalized concentration fluctuations under diffusion equilibrium. J. Appl Prob. 1, 47–68.CrossRefGoogle Scholar
Vere-Jones, D. (1966) Simple stochastic models for the release of quanta of transmitter from a nerve terminal. Aust. J. Statist. 8, 53–63.Google Scholar
Vere-Jones, D. (1968) Some applications of probability generating functionals to the study of input-output streams. J. R. Statist. Soc. B 30, 321–333.Google Scholar
Vere-Jones, D. (1970) Stochastic models for earthquake occurrence. J. R. Statist. Soc. B 32, 1–62.Google Scholar
Vere-Jones, D. (1971) The covariance measure of a weakly stationary random measure. Appendix to Daley (1971).Google Scholar
Westcott, M. (1971) On existence and mixing results for cluster point processes. J. R. Statist. B 33, 290–300.Google Scholar
Westcott, M. (1972) The probability generating functional. J. Aust. Math. Soc. (to appear).Google Scholar