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Weak limit results for the extremes of a class of shot noise processes

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Patrick Homble  and
William P. McCormick
Patrick Homble*
Affiliation:
University of Georgia
William P. McCormick*
Affiliation:
University of Georgia
*
∗Postal address: Department of Statistics, University of Georgia, Athens, GA 30602, USA.
∗Postal address: Department of Statistics, University of Georgia, Athens, GA 30602, USA.
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Abstract

Shot noise processes form an important class of stochastic processes modeling phenomena which occur as shocks to a system and with effects that diminish over time. In this paper we present extreme value results for two cases — a homogeneous Poisson process of shocks and a non-homogeneous Poisson process with periodic intensity function. Shocks occur with a random amplitude having either a gamma or Weibull density and dissipate via a compactly supported impulse response function. This work continues work of Hsing and Teugels (1989) and Doney and O'Brien (1991) to the case of random amplitudes.

Keywords

SHOT NOISEEXTREME VALUESSADDLEPOINT APPROXIMATION

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60F05: Central limit and other weak theorems 60G99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 3 , September 1995 , pp. 707 - 726
Copyright
Copyright © Applied Probability Trust 1995 

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References

Ballerini, R. (1995) Nonconstant levels for T-year return periods for two-dimensional Poisson processes with periodic intensity. Stat. Prob. Lett. 22, 43–47.CrossRefGoogle Scholar
Bingham, N. H., Goldie, C. H. and Teugels, J. L. (1986) Regular Variation. Cambridge University Press.Google Scholar
Cox, D. R. and Isham, V. (1980) Point Processes. Chapman and Hall, London.Google Scholar
Daley, D. J. and Vere-Jones, D. (1988) An Introduction to the Theory of Point Processes. Springer-Verlag, New York.Google Scholar
Daniels, H. E. (1954) Saddlepoint approximations in statistics. Ann. Math. Statist. 25, 631–649.Google Scholar
Davis, R. and Resnick, S. I. (1985) Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Prob. 13, 179–195.Google Scholar
Doney, R. A. and O'Brien, G. L. (1991) Loud shot noise. Ann. Appl. Prob. 1, 88–103.Google Scholar
Embrechts, P., Jensen, J. L., Maejima, M., Teugels, J. L. (1985) Approximations for compound Poisson and Pólya processes. Adv. Appl. Prob. 17, 623–637.Google Scholar
Homble, P. and Mccormick, W. P. (1993) Weak limit results for the extremes of a class of shot noise processes. TR#93-10. Dept. of Statistics. University of Georgia, Athens, GA.Google Scholar
Homble, P. and Mccormick, W. P. (1994) Extremes for shot noise processes with heavy tailed amplitudes. Preprint.Google Scholar
Hsing, T. L. and Teugels, J. L. (1989) Extremal properties of shot noise processes. Adv. Appl. Prob. 21, 513–525.Google Scholar
Hüsler, J. (1986) Extreme values of non-stationary random sequences. J. Appl. Prob. 23, 937–950.Google Scholar
Iranpour, R. and Chacon, P. (1988) Basic Stochastic Processes: Marc Kac Lectures. Macmillan, New York.Google Scholar
Jensen, J. L. (1988) Uniform saddlepoint approximations. Adv. Appl. Prob. 20, 622–634.Google Scholar
Jensen, J. L. (1991) Uniform saddlepoint approximations and log-concave densities. J. R. Statist. Soc. B53, 157–172.Google Scholar
Jensen, J. L. (1992) Saddlepoint approximations to the distribution of the total claim amount in some recent risk models. Scandinavian Actuarial J. To appear.Google Scholar
Matern, B. (1986) Spatial Variation. Lecture Notes in Statistics 36. Springer-Verlag, New York.Google Scholar
Parzen, E. (1962) Stochastic Processes. Holden Day, San Francisco.Google Scholar
Rodriguez-Iturbe, I., Cox, D. R. and Isham, V. (1987) Some models for rainfall based on stochastic point processes. Proc. R. Soc. London A410, 269–288.Google Scholar
Rodriguez-Iturbe, I., Cox, D. R. and Isham, V. (1988) A point process model for rainfall. Further developments. Proc. R. Soc. London A417, 283–298.Google Scholar
Rootzen, H. (1986) Extreme value theory for moving averages. Ann. Prob. 6, 847–869.Google Scholar
Vervaat, W. (1979) On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. Appl. Prob. 11, 750–783.Google Scholar