12 CL ´ EMENT DOMBRY, MICHAEL FALK, MAXIMILIAN ZOTT Note that (ii) implies the uniform integrability of the sequence ( X$_{n}$ )$_{n}$$_{∈}$ N . We first show (i). Obviously, G$_{n}$ ( f ) → ≀≀ f ≀≀$_{D}$ due to (7). Standard arguments such as the monotone convergence theorem yield G$_{n}$ ( f$_{n}$ ) → ≀≀ f ≀≀$_{D}$ if f$_{n}$, f ∈ ¯ C $^{−}$( S ) with ‖ f$_{n}$ − f ‖$_{∞}$ → 0. Now noticing that M ( n ) →$_{D}$ η , the assertion is immediate from the extended continuous mapping theorem, see cf. Billingsley (1968, Theorem 5.5). Now we proof (ii). Elementary calculations show that for all n ≥ 2 □ Corollary 3.2. Denote by M ( n ) := ∑ n i =1 1 { X ( i $^{)}$> max$_{1}$$_{≤}$$_{j<i}$ X ( j ) } the number of complete records among X $^{(1)}$, . . ., X ( n $^{)}$. Then Proof. The assertion follows from Theorem 3.1 and the fact that (∑ n i =1 a$_{i}$ i ) / log( n ) →$_{n}$$_{→∞}$ a , if ( a$_{n}$ )$_{n}$$_{∈}$ N is some real-valued sequence with a$_{n}$ →$_{n}$$_{→∞}$ a . □ The following lemma provides an alternative representation for the extremal concurrence probability. Denote by 1$_{A}$ the indicator function of some set A , i. e. 1$_{A}$( ω ) = 1, if ω ∈ A , and 1$_{A}$( ω ) = 0, else. Lemma 3.3. Let η = ( η$_{s}$ )$_{s}$$_{∈}$$_{S}$ be an SMSP in ¯ C $^{−}$( S ) with D -norm ‖·‖$_{D}$ and generator Z = ( Z$_{s}$ )$_{s}$$_{∈}$$_{S}$ , and f ∈ ¯ E $^{−}$( S ) . Then (i) E ( ≀≀ η ≀≀$_{D}$ ) = E ( ‖ 1 / Z ‖ − 1 D 1$_{{}$ Z > 0 } ) . E ( M ( n )) log( n ) →$_{n}$$_{→∞}$ E ( ≀≀ η ≀≀$_{D}$ ) . E ( X 2 n ) = ∫ ¯ C $^{−}$( S ) n $^{2}$P ( n ( U − 1) > f ) 2 ( P ∗ M ( n ) ) (d f ) ≤ ∫ ¯ C $^{−}$( S ) n $^{2}$P ( n ( U$_{s}$ − 1) > f ( s )) 2 ( P ∗ M ( n ) ) (d f ) = E ( M ( n ) s ) 2 ) = 2 n n + 1 ≤ 2 .