Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T16:44:31.275Z Has data issue: false hasContentIssue false
  • English
  • Français

The central limit theorem for the Poisson shot-noise process

Published online by Cambridge University Press:  14 July 2016

John A. Lane
John A. Lane*
Affiliation:
University College of Wales, Aberystwyth
*
∗ Postal address: Department of Statistics, University College of Wales, Penglais, Aberystwyth, SY23 3DB, Wales.
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

INFINITE DIVISIBILITYFELLER CANONICAL MEASUREASYMPTOTIC NORMALITYCLUSTER POINT PROCESSINTEGRATED SHOT-NOISE
Type
Research Papers
Information
Journal of Applied Probability , Volume 21 , Issue 2 , June 1984 , pp. 287 - 301
Copyright
Copyright © Applied Probability Trust 1984 

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.)

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, 264–296.Google Scholar
Daley, D. J. (1971) The definition of a multidimensional generalization of shot-noise. J. Appl. Prob. 8, 128–135.Google Scholar
Daley, D. J. (1972) Asymptotic properties of stationary point processes with generalized clusters. Z. Wahrscheinlichkeitsth. 21, 65–76.Google 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, 263–271.Google Scholar
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, 398–456.Google Scholar
Lewis, P. A. W. (1969) Asymptotic properties and equilibrium conditions for branching Poisson processes. J. Appl. Prob. 6, 355–371.Google 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, 480–485.Google Scholar
Pakes, A. G. (1972) A limit theorem for the integral of a critical age-dependent branching process. Math. Biosci. 13, 109–112.CrossRefGoogle Scholar
Parzén, E. (1962) Stochastic Processes. Holden-Day, San Francisco.Google Scholar
Rice, J. (1977) On generalized shot-noise. Adv. Appl. Prob. 9, 553–565.Google 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, 203–236.Google Scholar
Vere-Jones, D. (1970) Stochastic models for earthquake occurrence (with discussion). J. R. Statist. Soc. B 32, 1–62.Google Scholar
Westcott, M. (1973) Results in the asymptotic and equilibrium theory of Poisson cluster processes. J. Appl. Prob. 10, 807–823.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, 396–398.Google Scholar