On Functional Records and Champions

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,..., Xn-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,..., Xn-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.