The structure of extremal processes

Abstract

An extremal-Fprocess {Y(t);t≧ 0} is defined as the continuous time analogue of sample sequences of maxima of i.i.d. r.v.'s distributed likeFin the same way that processes with stationary independent increments (s.i.i.) are the continuous time analogue of sample sums of i.i.d. r.v.'s with an infinitely divisible distribution. Extremal-F processes are stochastically continuous Markov jump processes which traverse the interval of concentration ofF.Most extremal processes of interest are broad sense equivalent to the largest positive jump of a suitable s.i.i. process and this together with known results from the theory of record values enables one to conclude that the number of jumps ofY(t) in (t1,t2] follows a Poisson distribution with parameter logt2/t1. The time transformationt→etgives a new jump process whose jumps occur according to a homogeneous Poisson process of rate 1. This fact leads to information about the jump times and the inter-jump times. WhenFis an extreme value distribution theY-process has special properties. The most important is that ifF(x) = exp {—e–x} thenY(t) has an additive structure. This structure plus non parametric techniques permit a variety of conclusions about the limiting behaviour ofY(t) and its jump times.