  Consider $n$ i.i.d. random vectors on $\mathbb{R}^2$, with unknown, common
distribution function $F$. Under a sharpening of the extreme value condition on
$F$, we derive a weighted approximation of the corresponding tail copula
process. Then we construct a test to check whether the extreme value condition
holds by comparing two estimators of the limiting extreme value distribution,
one obtained from the tail copula process and the other obtained by first
estimating the spectral measure which is then used as a building block for the
limiting extreme value distribution. We derive the limiting distribution of the
test statistic from the aforementioned weighted approximation. This limiting
distribution contains unknown functional parameters. Therefore, we show that a
version with estimated parameters converges weakly to the true limiting
distribution. Based on this result, the finite sample properties of our testing
procedure are investigated through a simulation study. A real data application
is also presented.
