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The central limit theorem for the Poisson shot-noise process

Published online by Cambridge University Press:  14 July 2016

John A. Lane
Affiliation:
University College of Wales, Aberystwyth

Abstract

The Poisson shot-noise process discussed here takes the form f:oo H(t, s)N(ds), where N(· is the counting measure of a Poisson process and the H(·, s) are independent stochastic processes. Necessary and sufficient conditions are obtained for convergence in distribution, as t ∼ OC, to any infinitely divisible distribution. The main interest is in the explosive transient one-sided shot-noise, Y(t) = f:1 H(t, s)N(ds) where Var Y(t)∼ oc, Here conditions for asymptotic normality are discussed in detail. Important examples include the Poisson cluster point process and the integrated stationary shotnoise.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1984 

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Footnotes

The major part of this work was carried out while the author was a Ph.D. student in the Department of Mathematics, Imperial College of Science and Technology, London.

References

Bartlett, M. S. (1963) The spectral analysis of point processes (with discussion). J. R. Statist. Soc. B 25, 264296.Google Scholar
Daley, D. J. (1971) The definition of a multidimensional generalization of shot-noise. J. Appl. Prob. 8, 128135.CrossRefGoogle Scholar
Daley, D. J. (1972) Asymptotic properties of stationary point processes with generalized clusters. Z. Wahrscheinlichkeitsth. 21, 6576.CrossRefGoogle Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. Wiley, New York.Google Scholar
Jagers, P. (1967) Integrals of branching processes. Biometrika 54, 263271.CrossRefGoogle ScholarPubMed
Lane, J. A. (1979) Some Limit Theorems for Shot-noise Processes. , University of London.Google Scholar
Lewis, P. A. W. (1964) A branching Poisson process model for the analysis of computer failure patterns (with discussion). J. R. Statist. Soc. B 26, 398456.Google Scholar
Lewis, P. A. W. (1969) Asymptotic properties and equilibrium conditions for branching Poisson processes. J. Appl. Prob. 6, 355371.CrossRefGoogle Scholar
Lewis, P. A. W. (1970) Asymptotic properties of branching renewal processes. IBM Research Report RC2878, Yorktown Heights, N.Y. Google Scholar
Loève, M. (1963) Probability Theory, 3rd edn. Van Nostrand, Princeton, N.J. Google Scholar
Mcneil, D. R. (1970) Integral functionals of birth and death processes and related limiting distributions. Ann. Math. Statist. 41, 480485.CrossRefGoogle Scholar
Pakes, A. G. (1972) A limit theorem for the integral of a critical age-dependent branching process. Math. Biosci. 13, 109112.CrossRefGoogle Scholar
Parzén, E. (1962) Stochastic Processes. Holden-Day, San Francisco.Google Scholar
Rice, J. (1977) On generalized shot-noise. Adv. Appl. Prob. 9, 553565.CrossRefGoogle Scholar
Smith, W. L. (1964) Discussion contribution to Lewis, P. A. W. (1964). Google Scholar
Takács, L. (1954) On secondary processes generated by a Poisson process and their applications in physics. Acta Math. Acad. Sci. Hungar. 5, 203236.CrossRefGoogle Scholar
Vere-Jones, D. (1970) Stochastic models for earthquake occurrence (with discussion). J. R. Statist. Soc. B 32, 162.Google Scholar
Westcott, M. (1973) Results in the asymptotic and equilibrium theory of Poisson cluster processes. J. Appl. Prob. 10, 807823.CrossRefGoogle Scholar
Westcott, M. (1976) On the existence of a generalized shot-noise process. In Studies in Probability and Statistics: Papers in Honour of Edwin J. G. Pitman, ed. Williams, E. J., North-Holland, Amsterdam.Google Scholar
Westcott, M. (1977) A note on the non-homogeneous Poisson cluster process. J. Appl. Prob. 14, 396398.CrossRefGoogle Scholar

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