ON FUNCTIONAL RECORDS AND CHAMPIONS 11 where ≀≀ · ≀≀$_{D}$ is the dual D -norm function corresponding to ‖·‖$_{D}$ . We call E ( ≀≀ η ≀≀$_{D}$ ) the extremal concurrence probability corresponding to ‖·‖$_{D}$ , in accordance with the terminology in Dombry et al. (2015). As they have shown, the extremal concurrence probability has the following interpretation. It is well- known (cf. de Haan and Ferreira (2006, Corollary 9.4.2)), that the simple max- stable process ξ = − 1 / η has the representation where ( ϑ$_{k}$ )$_{k}$$_{∈}$ N are the points of a Poisson point process on (0 , ∞ ) × ¯ C + 1 ( S ) with a certain intensity measure. The extremal concurrence probability is now precisely the probability that only one function ϑ$_{k}$ contributes to the supremum in (11), see Dombry et al. (2015, Theorem 1). Note that one has to distinguish between E ( ≀≀ η ≀≀$_{D}$ ) and E (inf$_{s}$$_{∈}$$_{S}$ | η$_{s}$ | Z$_{s}$ ) in general. However, if η and Z are independent, both terms coincide, cf. Lemma 3.3. Proof of Theorem 3.1. Denote by P ∗ ξ the distribution of a random variable ξ . Let η be an SMSP with D -norm ‖·‖$_{D}$ and put M ( n ) := n max$_{i}$$_{=1}$$_{,...,n}$$_{−}$$_{1}$ ( U ( i ) − 1 ) →$_{D}$ η due to (5). Conditioning on M ( n ) = f yields since M ( n ) and U are independent. Setting X$_{n}$ := G$_{n}$ ◦ M ( n $^{)}$, we need to show It is enough to verify (Billingsley (1968, p. 32)): (i) X$_{n}$ →$_{D}$ ≀≀ η ≀≀$_{D}$ . (ii) There is ε > 0 with sup$_{n}$$_{∈}$$_{N}$ E ( | X$_{n}$ | 1+ ε ) < ∞ . n ¯ π$_{n}$ ( U ) = E ( X$_{n}$ ) →$_{n}$$_{→∞}$ = E ( ≀≀ η ≀≀$_{D}$ ) . n ¯ π$_{n}$ ( U ) = ∫ ¯ C $^{−}$( S ) nP ( n ( U − 1) > f ) ( P ∗ M ( n ) ) (d f ) =: ∫ ¯ C $^{−}$( S ) G$_{n}$ ( f ) ( P ∗ M ( n ) ) (d f ) (11) ξ =$_{D}$ sup k ∈ N ϑ$_{k}$ ,