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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: eliazar@math.biu.ac.il; eliazar@post.tau.ac.il
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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.

Information

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

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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