A pair-copula construction is a decomposition of a multivariate copula into a
structured system, called regular vine, of bivariate copulae or pair-copulae.
The standard practice is to model these pair-copulae parametrically, which
comes at the cost of a large model risk, with errors propagating throughout the
vine structure. The empirical pair-copula proposed in the paper provides a
nonparametric alternative still achieving the parametric convergence rate.

This paper suggests five measures of association between two random vectors X
= (X_1, ..., X_p) and Y = (Y_1, ..., Y_q). They are copula based and therefore
invariant with respect to the marginal distributions of the components X_i and
Y_j. The measures capture positive as well as negative association of X and Y.
In case p = q = 1 they reduce to Spearman's rho. Various properties of these
new measures are investigated. Nonparametric estimators, based on ranks, for
the measures are derived and their small sample behaviour is investigated by
simulation.

Starting from the characterization of extreme-value copulas based on
max-stability, large-sample tests of extreme-value dependence for multivariate
copulas are studied. The two key ingredients of the proposed tests are the
empirical copula of the data and a multiplier technique for obtaining
approximate p-values for the derived statistics. The asymptotic validity of the
multiplier approach is established, and the finite-sample performance of a
large number of candidate test statistics is studied through extensive Monte
Carlo experiments for data sets of dimension two to five.

Conditions are given under which the empirical copula process associated with
a random sample from a bivariate continuous distribution has a smaller
asymptotic covariance function than the standard empirical process based on
observations from the copula. Illustrations are provided and consequences for
inference are outlined.

Under an appropriate regular variation condition, the affinely normalized
partial sums of a sequence of independent and identically distributed random
variables converges weakly to a non-Gaussian stable random variable. A
functional version of this is known to be true as well, the limit process being
a stable L\'evy process.

The tail of a bivariate distribution function in the domain of attraction of
a bivariate extreme-value distribution may be approximated by the one of its
extreme-value attractor. The extreme-value attractor has margins that belong to
a three-parameter family and a dependence structure which is characterised by a
spectral measure, that is a probability measure on the unit interval with mean
equal to one half.

The quantification of diversification benefits due to risk aggregation plays
a prominent role in the (regulatory) capital management of large firms within
the financial industry. However, the complexity of today's risk landscape makes
a quantifiable reduction of risk concentration a challenging task. In the
present paper we discuss some of the issues that may arise. The theory of
second-order regular variation and second-order subexponentiality provides the
ideal methodological framework to derive second-order approximations for the
risk concentration and the diversification benefit.

Inference on an extreme-value copula usually proceeds via its Pickands
dependence function, which is a convex function on the unit simplex satisfying
certain inequality constraints. In the setting of an iid random sample from a
multivariate distribution with known margins and unknown extreme-value copula,
an extension of the Cap\'era\`a-Foug\`eres-Genest estimator was introduced by
D. Zhang, M. T. Wells and L. Peng [Journal of Multivariate Analysis 99 (2008)
577-588]. The joint asymptotic distribution of the estimator as a random
function on the simplex was not provided.

Consider a random sample from a bivariate distribution function $F$ in the
max-domain of attraction of an extreme-value distribution function $G$. This
$G$ is characterized by two extreme-value indices and a spectral measure, the
latter determining the tail dependence structure of $F$. A major issue in
multivariate extreme-value theory is the estimation of the spectral measure
$\Phi_p$ with respect to the $L_p$ norm.

Consider a continuous random pair $(X,Y)$ whose dependence is characterized
by an extreme-value copula with Pickands dependence function $A$. When the
marginal distributions of $X$ and $Y$ are known, several consistent estimators
of $A$ are available. Most of them are variants of the estimators due to
Pickands [Bull. Inst. Internat. Statist. 49 (1981) 859--878] and
Cap\'{e}ra\`{a}, Foug\`{e}res and Genest [Biometrika 84 (1997) 567--577]. In
this paper, rank-based versions of these estimators are proposed for the more
common case where the margins of $X$ and $Y$ are unknown.