  For multivariate distributions in the domain of attraction of a max-stable
distribution, the tail copula and the stable tail dependence function are
equivalent ways to capture the dependence in the upper tail. The empirical
versions of these functions are rank-based estimators whose inflated estimation
errors are known to converge weakly to a Gaussian process that is similar in
structure to the weak limit of the empirical copula process. We extend this
multivariate result to continuous functional data by establishing the
asymptotic normality of the estimators of the tail copula, uniformly over all
finite subsets of at most $D$ points ($D$ fixed). An application for testing
tail copula stationarity is presented. The main tool for deriving the result is
the uniform asymptotic normality of all the $D$-variate tail empirical
processes. The proof of the main result is non-standard.
