ON FUNCTIONAL RECORDS AND CHAMPIONS 19 Proof. Denote by U ( n ) the copula process corresponding to X ( n $^{)}$, n ∈ N . Taking logarithms, (14) becomes It can be shown by elementary arguments that (15) is equivalent to for each f ∈ E ( S ) with inf$_{s}$$_{∈}$$_{S}$ G$_{s}$ ( f ( s )) > 0. Hence, Lemma 3.9 and the strict monotonicity of F$_{s}$ entail □ Simple record probability. In the preceding section, we have investigated the (normalized) probability of a complete record and in particular, its limit, the ex- tremal concurrence probability. Now we will repeat this procedure, this time for the simple record probability. Unlike in the previous section, where we were actually dealing with the probability of having a champion, normalizing the record proba- bility with the factor n does not yield an interpretation in terms of a probability in the simple record case. The following result is the equivalent of Theorem 3.1 and Proposition 3.7 in the context of multivariate simple records. Let X , X$_{1}$ , X$_{2}$ , . . . be i. i d. rv in R d with (15) sup s ∈ S | n ( F$_{s}$ ( c$_{n}$ ( s ) x + d$_{n}$ ( s )) − 1) − log ( G$_{s}$ ( x )) | →$_{n}$$_{→∞}$ 0 . sup s ∈ S | ψ$_{n}$ ( f ( s )) − ψ ( f ( s )) | := sup s ∈ S | n ( F$_{s}$ ( c$_{n}$ ( s ) f ( s ) + d$_{n}$ ( s )) − 1) − log ( G$_{s}$ ( f ( s ))) | →$_{n}$$_{→∞}$ 0 nP ( X − d$_{n}$ c$_{n}$ > f, X > max i =1 ,...,n − $_{1}$X ( i ) ) = nP ( n ( U$_{s}$ − 1) > ψ$_{n}$ ( f ( s )) , s ∈ S, U > max i =1 ,...,n − $_{1}$U ( i ) ) →$_{n}$$_{→∞}$ E ( ≀≀ max( η , ψ ( f )) ≀≀$_{D}$ ) . 4. Simple records for multivariate observations