content_image/305470.jpg content_image/305469.jpg content_image/305468.jpg content_image/305467.jpg content_image/305465.jpg
given by 

<img src='content_image/305470.jpg'>

Hence, by (8), the attendant D -norm is for such x 

<img src='content_image/305469.jpg'>

Note that ‖·‖$_{α}$ is defined as in (9), even though it does not define a norm in the case α < 1. 

Example 2.7 (Bernoulli model) . A simple example of a discrete generator is in- duced by independent Bernoulli- β random variables ˜ Z$_{i}$ , i = 1 , . . . , d , β ∈ (0 , 1], and putting Z$_{i}$ := ˜ Z$_{i}$/β , i = 1 , . . . , d . The D -norm and the dual D -norm function are easily derived. We have 

<img src='content_image/305468.jpg'>

Note that ‖·‖$_{B}$$_{1}$ = ‖·‖$_{∞}$ and ‖·‖$_{B}$$_{β}$ → ‖·‖$_{1}$ as β → 0. Analogously, 

<img src='content_image/305467.jpg'>

## 3. The functional extremal concurrence probability 

The aim of this section is to investigate the limit behaviour of the sample concur- rence probability. In Dombry et al. (2015), it is shown that the sample concurrence probability p$_{n}$ ( X ) of a rv X converges, provided that X has continuous margins and lies in the max-domain of attraction of a max-stable rv. We generalize this assertion to the functional setup, having in mind that we can formulate every result in the multivariate context analogously. 

Theorem 3.1. Let U $^{(1)}$, U $^{(2)}$, . . . be independent copies of a copula process U , satisfying U ∈ D ( η ) , where η is an SMSP with corresponding D -norm ‖·‖$_{D}$ . Then 

<img src='content_image/305465.jpg'>

