content_image/298362.jpg content_image/298363.jpg content_image/298361.jpg content_image/298360.jpg
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 

<img src='content_image/298362.jpg'>

<img src='content_image/298363.jpg'>

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)) < ∞ . 

<img src='content_image/298361.jpg'>

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 

<img src='content_image/298360.jpg'>

