ON FUNCTIONAL RECORDS AND CHAMPIONS 29 Let X follow the df C . Next we show that E ( N (2)) is typically finite if X has at least two components X$_{j}$, X$_{k}$ which are tail independent, i. e. In case the limit exists, define the dependence measure where ( X$_{1}$, X$_{2}$ ) follows some bivariate copula, cf. Coles et al. (1999); Heffernan (2000). Note that we have ¯ χ = 1 if X$_{1}$, X$_{2}$ are tail dependent. In the class of (bivariate) copulas that are tail independent, however, ¯ χ is a popular measure of tail comparison. For a bivariate normal copula with coefficient of correlation ρ ∈ ( − 1 , 1) it is, for instance, well known that ¯ χ = ρ . Proposition 4.8. Let X = ( X$_{1}$, . . . , X$_{d}$ ) follow a copula C in R $^{d}$. Suppose that there exist indices k = j such that Then we have E ( N (2)) < ∞ . Corollary 4.9. We have E ( N (2)) < ∞ for multivariate normal rv unless all components are completely dependent. Proof of Proposition 4.8. We have where ∫ ∞ 1 P ( X$_{i}$ ≥ 1 − 1 t , 1 ≤ i ≤ d ) d t ≤ ∫ ∞ 1 P ( X$_{k}$ ≥ 1 − 1 t , X$_{j}$ ≥ 1 − 1 t ) d t = ∫ ∞ 1 exp ( log ( P ( X$_{k}$ ≥ 1 − 1 $_{t}$, X$_{j}$ ≥ 1 − 1 t )) log ( 1 t 2 ) log ( 1 t 2 ) d t, log ( P ( X$_{k}$ ≥ 1 − 1 $_{t}$, X$_{j}$ ≥ 1 − 1 t )) log ( 1 t 2 ) →$_{t}$$_{→∞}$ 1 1 + ¯ χ > 1 2 . ¯ χ$_{k,j}$ = lim u ↑ 1 2 log(1 − u ) log( P ( X$_{k}$ > u, X$_{j}$ > u )) − 1 ∈ ( − 1 , 1) . ¯ χ := lim u ↑ 1 2 log(1 − u ) log( P ( X$_{1}$ > u, X$_{2}$ > u )) − 1 ∈ [ − 1 , 1] , lim u ↑ $_{1}$P ( X$_{k}$ > u | X$_{j}$ > u ) = 0 .