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, wheretis the time at which the event took place andxis 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 thenth “runner-up” event, and the multidimensional distribution of the “topn” 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.