  The classical approach to multivariate extreme value modelling assumes that
the joint distribution belongs to a multivariate domain of attraction. This
requires each marginal distribution be individually attracted to a univariate
extreme value distribution. An apparently more flexible extremal model for
multivariate data was proposed by Heffernan and Tawn under which not all the
components are required to belong to an extremal domain of attraction but
assumes instead the existence of an asymptotic approximation to the conditional
distribution of the random vector given one of the components is extreme.
Combined with the knowledge that the conditioning component belongs to a
univariate domain of attraction, this leads to an approximation of the
probability of certain risk regions. The original focus on conditional
distributions had technical drawbacks but is natural in several contexts. We
place this approach in the context of the more general approach using
convergence of measures and multivariate regular variation on cones.
