  Multivariate extreme value theory assumes a multivariate domain of attraction
condition for the distribution of a random vector. This necessitates that each
component satisfies a marginal domain of attraction condition. An approximation
of the joint distribution of a random vector obtained by conditioning on one of
the components being extreme was developed by Heffernan and Tawn [12] and
further studied by Heffernan and Resnick [11]. These papers left unresolved the
consistency of different models obtained by conditioning on different
components being extreme and we here provide clarification of this issue. We
also clarify the relationship between these conditional distributions,
multivariate extreme value theory and standard regular variation on cones of
the form $[0,\infty]\times(0,\infty]$.
