  This article discusses modelling of the tail of a multivariate distribution
function by means of a large deviation principle (LDP), and its application to
the estimation of the probability of a multivariate extreme event from a sample
of n iid random vectors, with the probability bounded by powers of sample size
with exponents below -1. One way to view classical tail limits is as limits of
probability ratios. In contrast, the tail LDP provides asymptotic bounds or
limits for log-probability ratios. After standardising the marginals to
standard exponential, dependence is represented by a homogeneous rate function.
Furthermore, the tail LDP can be extended to represent both dependence and
marginals, the latter implying marginal log-GW tail limits. A connection is
established between the tail LDP and residual tail dependence (or hidden
regular variation) and a recent extension of it. Under a smoothness assumption,
they are implied by the tail LDP. Based on the tail LDP, a simple estimator for
very small probabilities of extreme events is formulated. It avoids estimation
of the rate function by making use of its homogeneity. Strong consistency in
the sense of convergence of log-probability ratios is proven. Simulations and
an application illustrate the difference between the classical approach and the
LDP-based approach.
