15 ON FUNCTIONAL RECORDS AND CHAMPIONS The above remark shows that we do not have to limit our considerations to copula processes. If, for instance, X is an MSP itself with univariate marginal distributions G$_{s}$ , s ∈ S , then η := (log ( G$_{s}$ ( X$_{s}$ )))$_{s}$$_{∈}$$_{S}$ is an SMSP. Applying the max-stability of η , we obtain where ‖·‖$_{D}$ is the D -norm corresponding to η , and η $^{(1)}$, η $^{(2)}$,... are iid copies of η . Having established the functional extremal concurrence probability, we can now derive the limit survival function of a complete record. We will have to restrict to the case where P ( Z > 0) > 0, which is equivalent to the fact that the extremal concurrence probability is positive, cf. (12). Just like before, we consider the copula process case first. Proposition 3.7. In addition to the assumptions of Theorem 3.1, suppose that the generator fulfills P ( Z > 0) > 0 . Then, for f ∈ ¯ E $^{−}$( S ) , where η = ( η$_{s}$ )$_{s}$$_{∈}$$_{S}$ is an SMSP with corresponding D -norm ‖·‖$_{D}$ . Note that we avoid division by zero in the preceding formula since we assume P ( Z > 0) > 0. P ( n ( U ( n ) − 1 ) > f ∣ U ( n ) is a complete record ) =: ¯ H$_{n}$ ( f ) →$_{n}$$_{→∞}$ ¯ H$_{D}$ ( f ) := E ( ≀≀ max( η ,f ) ≀≀$_{D}$ ) E ( ≀≀ η ≀≀$_{D}$ ) , ¯ π$_{n}$ ( X ) = ¯ π$_{n}$ ( η ) = P ( η > max i =1 ,...,n − $_{1}$η ( i ) ) = P ( ( n − 1) η > η (1) ) = ∫ ¯ C $^{−}$( S ) P ( n − 1) f > η (1) ) ( P ∗ η )(d f ) = ∫ ¯ C $^{−}$( S ) exp ( − ( n − 1) ‖ f ‖$_{D}$ ) ( P ∗ η )(d f ) = E (exp ( − ( n − 1) ‖ η ‖$_{D}$ )) ,