arXiv:1510.04529v1 [math.PR] 15 Oct 2015 2010 Mathematics Subject Classification. Primary 60G70. Key words and phrases. Champions and records • multivariate extreme value distribution • max- stable random vectors • D -norm • max-stable processes • max-domain of attraction. This work was supported by a research grant (VKR023480) for the second author from VILLUM FONDEN. 1 (1) max i =1 ,...,n ϑ ( i ) − b$_{n}$ a$_{n}$ =$_{D}$ ϑ , ON FUNCTIONAL RECORDS AND CHAMPIONS CL ´ EMENT DOMBRY, MICHAEL FALK, MAXIMILIAN ZOTT Abstract. Records among a sequence of iid random variables X$_{1}$, X$_{2}$, . . . on the real line have been investigated extensively over the past decades. A record is defined as a random variable X n such that X n > max( X$_{1}$, . . . , X n − $_{1}$). Trying to generalize this concept to the case of random vectors, or even sto- chastic processes with continuous sample paths, the question arises how to define records in higher dimensions. We introduce two different concepts: A simple record is meant to be 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. The behavior of records is investigated. 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 distribution of X n , given that X n is a record is derived. Preliminaries. Let S be a compact metric space. A max-stable process (MSP) ϑ = ( ϑ$_{s}$ )$_{s}$$_{∈}$$_{S}$ is a stochastic process with non-degenerate univariate margins and sample paths in C ( S ) := { f ∈ R S : f continuous } with the property that there are functions a$_{n}$ ∈ C $^{+}$( S ) := { f ∈ C ( S ) : f > 0 } , b$_{n}$ ∈ C ( S ), n ∈ N , such that where ϑ $^{(1)}$, . . . , ϑ ( n ) are independent and identically distributed (iid) copies of ϑ 1. Introduction and Preliminaries