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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.

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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