Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-13T00:45:33.377Z Has data issue: false hasContentIssue false
  • English
  • Français

Extremal properties of shot noise processes

Published online by Cambridge University Press:  01 July 2016

Tailen Hsing  and
J. L. Teugels
Tailen Hsing*
Affiliation:
Texas A&M University
J. L. Teugels*
Affiliation:
Katholieke University Leuven
*
∗Postal address: Department of Statistics, Texas A&M University, College Station, TX 77843–3143, USA.
∗∗Postal address: Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3030 Heverlee, Belgium.
Get access Rights & Permissions [Opens in a new window]

Abstract

Consider the shot noise process X(t):= Σih(t – τi), , where h is a bounded positive non-increasing function supported on a finite interval, and the are the points of a renewal process η on [0, ). In this paper, the extremal properties of {X(t)} are studied. It is shown that these properties can be investigated in a natural way through a discrete-time process which records the states of {X(t)} at the points of η. The important special case where η is Poisson is treated in detail, and a domain-of-attraction result for the compound Poisson distribution is obtained as a by-product.

Keywords

EXTREME VALUEPOINT PROCESSRENEWAL PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 21 , Issue 3 , September 1989 , pp. 513 - 525
Copyright
Copyright © Applied Probability Trust 1989 

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

[1] Bingham, N. H., Goldie, C. H. and Teugels, J. L. (1986) Regular Variation. Cambridge University Press.Google Scholar
[2] Conference on Physical Aspects of Noise in Electronic Devices (1968) Peter Peregrinus, Stevenage.Google Scholar
[3] Daley, D. J. (1971) The definition of multi-dimensional generalization of shot noise. J. Appl. Prob. 8, 128–135.Google Scholar
[4] De Haan, L. (1970) On Regular Variation and Its Applications to the Weak Convergence of Sample Extremes. Mathematical Center.Google Scholar
[5] Dogliotti, R., Luvison, A. and Puirani, G. (1979) Error probability in optical fiber transmission systems. IEEE Trans. Inf. Theory 25, 170–178.Google Scholar
[6] Embrechts, P., Jensen, J. L., Maejima, M. and Teugels, J. L. (1985) Approximations for compound Poisson and Polya processes. Adv. Appl. Prob. 17, 623–637.Google Scholar
[7] Feigin, P. D. and Yashchin, E. (1983) On a strong Tauberian result. Z. Wahrscheinlichkeitsth. 65, 35–48.Google Scholar
[8] Galambos, J. (1978) The Asymptotic Theory of Extreme Order Statistics. Wiley, New York.Google Scholar
[9] Gilbert, E. N. and Pollack, H. O. (1960) Amplitude distribution of shot noise. Bell System Tech. J. 30, 333–350.Google Scholar
[10] Hsing, T., Husler, J. and Leadbetter, M. R. (1985) On the exceedance process for a stationary sequence. Prob. Theory. Rel. Fields 78, 97–112.CrossRefGoogle Scholar
[11] Jensen, J. L. (1988) Uniform saddlepoint approximations. Adv. Appl. Prob. 20, 622–634.CrossRefGoogle Scholar
[12] Kallenberg, O. (1982) Random Measures. Akademie-Verlag, Berlin: Academic Press, New York.Google Scholar
[13] Kuno, A. and Ikegaya, K. (1973) A statistical investigation of acoustic power radiated by a flow of random point sources. J. Acoust. Soc. Japan 29, 662–671.Google Scholar
[14] Leadbetter, M. R., Lindgren, G. and Rootzén, H. (1983) Extremes and Related Propeties of Random Sequences and Processes. Springer-Verlag, New York.Google Scholar
[15] Lugannani, R. (1978) Sample functions regularity of shot processes. SIAM J. Appl. Math. 35, 249–259.CrossRefGoogle Scholar
[16] Middleton, D. (1973) Man-made noise in urban environments and transportation systems: Models and measurements. IEEE Trans. Comm. 21, 1232–1241.Google Scholar
[17] Papoulis, A. (1971) High density shot noise and Gaussianity. J. Appl. Prob. 8, 1181–127.Google Scholar
[18] Parzen, E. (1962) Stochastic Processes. Holden-Day, San Francisco.Google Scholar
[19] Reed, M. and Simon, B. (1975) Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis, Self-Adjointness. Academic Press, New York.Google Scholar
[20] Rice, J. (1977) On generalized shot noise. Adv. Appl. Prob. 9, 553–565.Google Scholar
[21] Rice, S. O. (1944) Mathematical analysis of random noise. Bell System Tech. J. 23, 282–332.Google Scholar
[22] Rootzén, H. (1978) Extremes of moving averages of stable processes. Ann. Prob. 6, 847–869.Google Scholar
[23] Rootzen, H. (1988) Maxima and exceedances of stationary Markov chains. Adv. Appl. Prob. 20, 371–390.Google Scholar
[24] Tung, C. C. (1967) Random response of highway bridges to vehicle loads. J. Engrg. Mech. Div., Proc. Amer. Soc. Civil Engineers EM1 93, 79–94.Google Scholar
[25] Verveen, A. and Defelice, L. (1974) Membrane noise. Prog. Biophys. Mol. Biol. 28, 189–265.Google Scholar
[26] Westcott, M. (1976) On the existence of a generalized shot-noise process. Studies in Probability and Statistics. North-Holland, Amsterdam.Google Scholar
[27] Yarovaya, N. V. (1983) Some properties of shot-effect fields. Theory Prob. Math. Statist. 27, 167–173.Google Scholar