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Negative binomial processes

Published online by Cambridge University Press:  14 July 2016

Ole Barndorff-Nielsen  and
G. F. Yeo
Ole Barndorff-Nielsen
Affiliation:
Aarhus University
G. F. Yeo
Affiliation:
Aarhus University
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Summary

This paper is concerned with negative binomial processes which are essentially mixed Poisson processes whose intensity parameter is given by the sum of squares of a finite number of independently and identically distributed Gaussian processes. A study is made of the distribution of the number of points of a k-dimensional negative binomial process in a compact subset of Rk, and in particular in the case where the underlying Gaussian processes are independent Ornstein-Uhlenbeck processes when more detailed results may be obtained.

Type
Research Papers
Information
Journal of Applied Probability , Volume 6 , Issue 3 , December 1969 , pp. 633 - 647
Copyright
Copyright © Applied Probability Trust 

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References

Anscombe, F. (1950) Sampling theory of the negative binomial and logarithmic series distributions. Biometrika 37, 358–382.Google Scholar
Bartlett, M. S. (1960) Stochastic Population Models in Ecology and Epidemiology. Methuen, London.Google Scholar
Bartlett, M. S. (1964) The spectral analysis of two-dimensional point processes. Biometrika 51, 299–311.Google Scholar
Bates, G. E. and Neyman, J. (1952) Contributions to the theory of accident proneness. University of California Publications in Statistics 1, 215–276.Google Scholar
Cox, D. R. and Lewis, P. A. W. (1966) The Statistical Analysis of a Series of Events. Methuen, London.Google Scholar
Darling, D. A. and Siegert, A. J. F. (1957) A systematic approach to a class of problems in the theory of noise and other random phenomena: Part I. IRE Transactions on Information Theory, IT–3, 32–36.Google Scholar
Dunford, N. and Schwartz, J. T. (1963) Linear Operators. Part II: Spectral Theory. Interscience, New York.Google Scholar
Greenwood, M. and Yule, G. (1920) An inquiry into the nature of frequency distributions of multiple happenings etc. J. R. Statist. Soc. 83, 255–279.Google Scholar
Kac, M. and Siegert, A. J. F. (1947) An explicit representation of a stationary Gaussian process. Ann. Math. Statist. 18, 438–442.Google Scholar
Kalinin, V. M. (1967) Convergent and asymptotic expansions for probability distributions. Theor. Probability Appl. 12, 22–35.Google Scholar
Krishnaiah, P. R. and Rao, M. M. (1961) Remarks on a multivariate gamma distribution. Amer. Math. Monthly 68, 342–346.Google Scholar
Krishnamoorthy, A. S. and Parthasarathy, M. (1951) A multivariate gamma-type distribution. Ann. Math. Statist. 22, 549–557.Google Scholar
Lukacs, E. and Laha, R. G. (1964) Applications of Characteristic Functions. Griffin, London.Google Scholar
Lundberg, O. (1940) On Random Processes and Their Applications to Sickness and Accident Statistics. Thesis, Uppsala.Google Scholar
Mcfadden, J. A. (1965) The mixed Poisson process. Sankhya., A 27, 83–92 Google Scholar
Middleton, D. (1960) An Introduction to Statistical Communication Theory. McGraw-Hill, New York.Google Scholar
Neyman, J. (1965) Certain chance mechanisms involving discrete distributions. Classical and Contagious Discrete Distributions. Pergamon Press.Google Scholar
Neyman, J. and Scott, E. L. (1958) On a mathematical theory of populations conceived as conglomerations of clusters. Cold Spring Harbour Symposium, 109–120.Google Scholar
Siegert, A. J. F. (1957) A systematic approach to a class of problems in the theory of noise and other random phenomena: Part II. IRE Transactions on Information Theory, IT–3, 37–43.Google Scholar
Smithies, F. (1962) Integral Equations. Cambridge University Press.Google Scholar
Taylor, L. R. (1961) Aggregation, variance and the mean. Nature 189, 732–735.Google Scholar
Vere-Jones, D. (1967) The infinite divisibility of a bivariate gamma distribution. Sankhya A 29, 421–422.Google Scholar
Williamson, E. and Bretherton, M. H. (1963) Tables of the negative binomial probability distribution. Wiley, New York.Google Scholar
ørESUNDS-VAND-KOMITEENS UNDERSøgELSER, 1959–1967. (1967) Copenhagen.Google Scholar