  Models based on assumptions of multivariate regular variation and hidden
regular variation provide ways to describe a broad range of extremal dependence
structures when marginal distributions are heavy tailed. Multivariate regular
variation provides a rich description of extremal dependence in the case of
asymptotic dependence, but fails to distinguish between exact independence and
asymptotic independence. Hidden regular variation addresses this problem by
requiring components of the random vector to be simultaneously large but on a
smaller scale than the scale for the marginal distributions. In doing so,
hidden regular variation typically restricts attention to that part of the
probability space where all variables are simultaneously large. However, since
under asymptotic independence the largest values do not occur in the same
observation, the region where variables are simultaneously large may not be of
primary interest. A different philosophy was offered in the paper of Heffernan
and Tawn [J. R. Stat. Soc. Ser. B Stat. Methodol. 66 (2004) 497--546] which
allows examination of distributional tails other than the joint tail. This
approach used an asymptotic argument which conditions on one component of the
random vector and finds the limiting conditional distribution of the remaining
components as the conditioning variable becomes large. In this paper, we
provide a thorough mathematical examination of the limiting arguments building
on the orientation of Heffernan and Tawn [J. R. Stat. Soc. Ser. B Stat.
Methodol. 66 (2004) 497--546]. We examine the conditions required for the
assumptions made by the conditioning approach to hold, and highlight
simililarities and differences between the new and established methods.
