The structure of extremal processes

Abstract

An extremal- F process { Y ( t ); t ≧ 0} is defined as the continuous time analogue of sample sequences of maxima of i.i.d. r.v.'s distributed like F in 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 of F. 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 of Y ( t ) in ( t 1 , t 2 ] follows a Poisson distribution with parameter log t 2 / t 1 . The time transformation t → e t gives 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. When F is an extreme value distribution the Y -process has special properties. The most important is that if F ( x ) = exp {— e – x } then Y ( t ) has an additive structure. This structure plus non parametric techniques permit a variety of conclusions about the limiting behaviour of Y ( t ) and its jump times.