Limit Laws for Random
  Vectors with an Extreme Component

<br>
<br>
 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 \cite{heffernan:tawn:2004}
    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
    \cite{heffernan:tawn:2004}.  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.