Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-25T08:09:22.819Z Has data issue: false hasContentIssue false
  • English
  • Français

Shot noise on cluster processes with cluster marks, and studies of long range dependence

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Filemon Ramirez-Perez  and
Robert Serfling
Filemon Ramirez-Perez*
Affiliation:
Universidad Autonoma Chapingo
Robert Serfling*
Affiliation:
University of Texas at Dallas
*
∗ Postal address: Area de Estadistica, Departamento de Fitotecnia, Universidad Autonoma Chapingo, Chapingo Edo. Mex. CP56230, Mexico.
∗∗ Postal address: Department of Mathematical Sciences, University of Texas at Dallas, Richardson, Texas 75083-0688, USA. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

With the aim of providing greater flexibility in developing and applying shot noise models, this paper studies shot noise on cluster point processes with both pointwise and cluster marks. For example, in financial modelling, responses to events in the financial market may occur in clusters, with random amplitudes including a ‘cluster component’ reflecting a degree of commonness among responses within a cluster. For such shot noise models, general formulae for the characteristic functional are developed and specialized to the case of Neyman-Scott clustering with cluster marks. For several general forms of response function, long range dependence of the corresponding equilibrium shot noise models is investigated. It is shown, for example, that long range dependence holds when the ‘structure component’ of the response function decays slowly enough, or when the response function has a finite random duration with a heavy tailed distribution.

Keywords

Shot noisepoint processescluster processeslong range dependenceheavy tailsequilibrium behaviourfinancial modelling

MSC classification

Primary: 60G99: None of the above, but in this section
Secondary: 60G35: Signal detection and filtering
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 33 , Issue 3 , September 2001 , pp. 631 - 651
Copyright
Copyright © Applied Probability Trust 2001 

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

Supported by NSF Grants DMS-9705209 and DMS-0103698.

References

Beran, J. (1994). Statistics for Long Memory Processes. Chapman and Hall, London.Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press.CrossRefGoogle Scholar
Bondesson, L. (1988). Shot-noise processes. In Encyclopedia of Statistical Science, Vol. 8, eds Kotz, S. et al. John Wiley, New York, pp. 448–452.Google Scholar
Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd edn. Springer, Berlin.CrossRefGoogle Scholar
Bruneel, H. and Kim, B. G. (1993). Discrete-Time Models for Communication Systems including ATM. Kluwer, Dordrecht.CrossRefGoogle 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 Intoduction to the Theory of Point Processes. Springer, Berlin.Google Scholar
Gilchrist, J. H. and Thomas, J. B. (1975). A shot process with burst properties. Adv. Appl. Prob. 7, 527–541.CrossRefGoogle Scholar
Heinrich, L. and Schmidt, V. (1985). Normal convergence of multidimensional shot noise and rates of this convergence. Adv. Appl. Prob. 17, 709–730.CrossRefGoogle Scholar
Hsing, T. and Teugels, J. L. (1989). Extremal properties of shot noise processes. Adv. Appl. Prob. 21, 513–525.CrossRefGoogle Scholar
Lane, J. A. (1984). The central limit theorem for the Poisson shot-noise process. J. Appl. Prob. 21, 287–301.CrossRefGoogle Scholar
Lane, J. A. (1987). The Berry–Esseen bound for the Poisson shot-noise. Adv. Appl. Prob. 19, 512–514.CrossRefGoogle Scholar
Lowen, S. B. and Teich, M. C. (1990). Power-law shot noise. IEEE Trans. Inf. Theory 36, 1302–1318.CrossRefGoogle Scholar
Parzen, E. (1962). Stochastic Processes. Holden-Day, San Francisco.Google Scholar
Ramirez-Perez, F. and Serfling, R. (2001). Asymptotic normality of shot noise on Poisson cluster processes with cluster marks. Submitted.Google Scholar
Resnick, S. (1997). Heavy tail modeling and teletraffic data (including discussion). Ann. Statist. 25, 1805–1866.CrossRefGoogle Scholar
Rice, J. (1977). On generalized shot noise. Adv. Appl. Prob. 9, 553–565.CrossRefGoogle Scholar
Samorodnitsky, G. (1995). A class of shot noise models for financial applications. In Proc. Athens Int. Conf. Appl. Prob. and Time Series, Vol. 1, eds Heyde, C. C., Prohorov, Y. V., Pyke, R. and Rachev, S. T.. Springer, Berlin, pp. 332–353.Google Scholar
Samorodnitsky, G. (1998). Tail behavior of some shot noise processes. In A Practical Guide to Heavy Tails: Statistical Techniques and Applications, eds Adler, R. J., Feldman, R. E. and Taqqu, M. S.. Birkhäuser, Basel, pp. 473–478.Google Scholar
Snyder, D. L. and Miller, M. I. (1991). Random Point Processes in Time and Space, 2nd edn. Springer, Berlin.CrossRefGoogle Scholar