On Functional Records and Champions
Résumé
A record among a sequence of iid random variables X 1 , X 2 ,. .. on the real line is defined as a member X n such that X n > max(X 1 ,. .. , X n−1). Trying to generalize this concept to random vectors, or even stochastic processes with continuous sample paths, we introduce two different concepts: A simple record is a stochastic process (or a random vector) X n that is larger than X 1 ,. .. , X n−1 in at least one component, whereas a complete record has to be larger than its predecessors in all components. In particular, the probability that a stochastic process X n is a record as n tends to infinity is studied, assuming that the processes are in the max-domain of attraction of a max-stable process. Furthermore, the conditional distribution of X n given that X n is a record is derived. Keywords Champions and records • Multivariate extreme value distribution • Max-stable random vectors • D-norm • Max-stable processes • Max-domain of attraction