EXTREMES: A CONTINUOUS-TIME PERSPECTIVE

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 n th “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.