Hostname: page-component-77c89778f8-cnmwb Total loading time: 0 Render date: 2024-07-16T16:43:22.994Z Has data issue: false hasContentIssue false
  • English
  • Français

Limiting properties of Poisson shot noise processes

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Robert Lund ,
William P. McCormick  and
Yuanhui Xiao
Robert Lund*
Affiliation:
University of Georgia
William P. McCormick*
Affiliation:
University of Georgia
Yuanhui Xiao*
Affiliation:
University of Rochester
*
∗ Postal address: Department of Statistics, University of Georgia, Athens, GA 30602-1952, USA
∗ Postal address: Department of Statistics, University of Georgia, Athens, GA 30602-1952, USA
∗ Postal address: Department of Biostatistics and Computational Biology, University of Rochester, Rochester, NY 14642, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper studies limiting properties of discretely sampled Poisson shot noise processes. Versions of the law of large numbers and central limit theorem are derived under very general conditions. Examples demonstrating the utility of the results are included.

Keywords

Central limit theoremcompound Poisson processimpulse responselaw of large numbersshot noise

MSC classification

Primary: 60G25: Prediction theory
Secondary: 60G20: Generalized stochastic processes
Type
Short Communications
Information
Journal of Applied Probability , Volume 41 , Issue 3 , September 2004 , pp. 911 - 918
Copyright
Copyright © Applied Probability Trust 2004 

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.)

References

Bevan, S., Kullberg, R., and Rice, J. (1979). An analysis of membrane noise. Ann. Statist. 7, 237–257.10.1214/aos/1176344609Google Scholar
Brockwell, P. J., Resnick, S. I., and Tweedie, R. L. (1982). Storage processes with general release and additive inputs, Adv. Appl. Prob. 14, 392–433.10.2307/1426528Google Scholar
Doney, R. A., and O'Brien, G. L. (1991). Loud shot noise. Ann. Appl. Prob. 1, 88–103.10.1214/aoap/1177005982Google Scholar
Hall, P., and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York.Google Scholar
Heinrich, L., and Schmidt, V. (1985). Normal convergence of multidimensional shot noise and rates of this convergence. Adv. Appl. Prob. 17, 709–730.10.2307/1427084Google Scholar
Īl'enko, A. B. (2002a). Asymptotic behavior of sample means of shot noise processes. Theory Prob. Math. Statist. 64, 57–65.Google Scholar
Īl'enko, A. B. (2002b). On the limit distribution of integrals of shot-noise processes. Ukrainian Math. J. 54, 64–75.10.1023/A:1019741519340Google Scholar
Lane, J. A. (1984). The central limit theorem for the Poisson-shot noise process. J. Appl. Prob. 21, 287–301.10.2307/3213640Google Scholar
Lane, J. A. (1987). The Berry-Eseen bound for the Poisson-shot noise. Adv. Appl. Prob. 19, 512–514.10.2307/1427432Google Scholar
Lauger, P. (1975). Shot noise in ion channels. Biochem. Biophys. Acta 413, 1–10.10.1016/0005-2736(75)90053-XGoogle Scholar
Lund, R. B., Butler, R., and Paige, R. L. (1999). Prediction of shot noise. J. Appl. Prob. 36, 374–388.10.1239/jap/1032374459Google Scholar
McCormick, W. P. (1998). Extremes for shot noise processes with heavy tailed amplitudes. J. Appl. Prob. 34, 643–656.10.2307/3215091Google Scholar
Moran, P. A. P. (1969). Dams in series with continuous release. J. Appl. Prob. 4, 380–388.10.2307/3212031Google Scholar
Papoulis, A. (1971). A high density shot noise and Gaussianity. J. Appl. Prob. 8, 118–127.10.2307/3211842Google Scholar
Parzen, E. (1962). Stochastic Processes. Holden Day, San Francisco, CA.Google Scholar
Perry, D., Stadje, W., and Zacks, S. (2001). First-exit times for Poisson shot noise. Stoch. Models 17, 25–37.10.1081/STM-100001398Google Scholar
Ramirez-Perez, F., and Serfling, R. (2001). Shot noise on cluster processes with cluster marks, and studies of long range dependence. Adv. Appl. Prob. 33, 631–651.10.1239/aap/1005091357Google Scholar
Ramirez-Perez, F., and Serfling, R. (2003). Asymptotic normality of shot noise on Poisson cluster processes with cluster marks. J. Prob. Statist. Sci. 1, 157–172.Google Scholar
Rice, J. (1977). On generalized shot noise. Adv. Appl. Prob. 9, 553–565.10.2307/1426114Google Scholar
Ross, S. (1996). Stochastic Processes, 2nd edn. John Wiley, New York.Google Scholar
Samorodnitsky, G. (1997). A class of shot-noise models for financial applications. In Athens Conference on Applied Probability and Time Series, Vol. 1, eds Heyde, C. C., Prohorov, Yu. 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