On Functional Records and Champions

Abstract

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