On Functional Records and Champions

A record among a sequence of iid random variables $$X_1,X_2,\dots $$ X 1 , X 2 , ⋯ on the real line is defined as a member $$X_n$$ X n such that $$X_n>\max (X_1,\cdots ,X_{n-1})$$ 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) $${\varvec{X}}_n$$ X n that is larger than $${\varvec{X}}_1,\cdots ,{\varvec{X}}_{n-1}$$ 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 $${\varvec{X}}_n$$ 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 $${\varvec{X}}_n$$ X n given that $${\varvec{X}}_n$$ X n is a record is derived.