Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T08:45:55.920Z Has data issue: false hasContentIssue false
  • English
  • Français

EXTREMES: A CONTINUOUS-TIME PERSPECTIVE

Published online by Cambridge University Press:  22 June 2005

Iddo Eliazar
Iddo Eliazar
Affiliation:
Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel, E-mail: [email protected]; [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a generic continuous-time system in which events of random magnitudes occur stochastically and study the system's extreme-value statistics. An event is described by a pair (t,x) of coordinates, where t is the time at which the event took place and x is the magnitude of the event. The stochastic occurrence of the events is assumed to be governed by a Poisson point process.

We study various issues regarding the system's extreme-value statistics, including (i) the distribution of the largest-magnitude event, the distribution of the nth “runner-up” event, and the multidimensional distribution of the “top n” extreme events, (ii) the internal hierarchy of the extreme-value events—how large are their magnitudes when measured relative to each other, and (iii) the occurrence of record times and record values. Furthermore, we unveil a hidden Poissonian structure underlying the system's sequence of order statistics (the largest-magnitude event, the second largest event, etc.). This structure provides us with a markedly simple simulation algorithm for the entire sequence of order statistics.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 19 , Issue 3 , July 2005 , pp. 289 - 308
Copyright
© 2005 Cambridge University Press

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

REFERENCES

Bertoin, J. (1996). Lévy processes. Cambridge: Cambridge University Press. Bertoin, J. (1999). Subordinators: Examples and applications. Lecture Notes in Mathematics 1717. New York: Springer-Verlag.
Eliazar, I. (2005). On geometric record times. Physica A 348: 181–198.Google Scholar
Embrechts, P., Kluppelberg, C., & Mikosch, T. (1997). Modelling extremal events for insurance and finance. Springer-Verlag.
Embrechts, P. & Omey, E. (1983). On subordinated distributions and record processes. Mathematical Proceedings of the Cambridge Philosophical Society 93: 339–353.Google Scholar
Evans, M., Hastings, N., & Peacock, B. (1993). Statistical distributions, 2nd ed. New York: Wiley.
Feller, W. (1971). An introduction to probability theory and its applications, Vol. 2, 2nd ed. New York: Wiley.
Fisher, R.A. & Tippett, L.H.C. (1928). Limiting forms of the frequency distribution of the largest and smallest member of a sample. Proceedings of the Cambridge Philosophical Society 24: 180–190.Google Scholar
Fréchet, M. (1927). Sur la loi de probabilité de l'écart maximum. Ann. Soc. Polon. Math. Cracovie 6: 93–116.Google Scholar
Galambos, J. (1987). Asymptotic theory of extreme order statistics, 2nd ed. New York: Krieger.
Gaver, D.P. (1976). Random record models. Journal of Applied Probability 13: 538–547.Google Scholar
Gnedenko, B. (1943). Sur la distribution limite du terme maximum d'une serie aleatoire. Annals of Mathematics 44: 423. Translated and reprinted in S. Kotz & N.L. Johnson, eds. (1992). Breakthroughs in statistics I. Berlin: Springer-Verlag, pp. 195–225.Google Scholar
Gumbel, E.J. (1958). Statistics of extremes. New York: Columbia University Press.
Ibragimov, I.A. & Linnik, Yu.V. (1971). Independent and stationary sequences of random variables. Groningen: Walters-Noordhoff.
Kingman, J.F.C. (1993). Poisson processes. Oxford: Oxford University Press.
Kotz, S. & Nadarajah, S. (2000). Extreme value distributions. London: Imperial College Press.
Levikson, B., Rolski, T., & Weiss, G. (1999). On the Poisson hyperbolic staircase. Probability in the Engineering and Informational Sciences 13: 11–32.Google Scholar
Reiss, R.D. (1993). A course on point processes. New York: Springer-Verlag.
Rényi, A. (1976). On outstanding values of a sequence of observations. In: P. Turan (ed.), Selected papers of A. Rényi, Vol. 3. Budapest: Akadémiai Kiadó.
Thomas, M. & Reiss, R.D. (2001). Statistical analysis of extreme values. Boston: Birkhauser.
von Bortkiewicz, L. (1922). Variationsberiteund mittlerer Fehler, Sitzungsber. Berlin Math. Ges. 21: 3.Google Scholar
von Mises, R. (1936). La distribution de la plus grande de n valuers. Rev. Math. Union Interbalcanique 1: 141–160. Translated and reprinted in (1964). Selected papers of Richard von Mises II. Providence, RI: American Mathematical Society, pp. 271–294.Google Scholar
Westcott, M. (1977). The random record model. Proceedings of the Royal Society, London, A 356: 529.Google Scholar