  The problem of inferring the distribution of a random vector given that its
norm is large requires modeling a homogeneous limiting density. We suggest an
approach based on graphical models which is suitable for high-dimensional
vectors.
  We introduce the notion of one-component regular variation to describe a
function that is regularly varying in its first component. We extend the
representation and Karamata's theorem to one-component regularly varying
functions, probability distributions and densities, and explain why these
results are fundamental in multivariate extreme-value theory. We then
generalize Hammersley-Clifford theorem to relate asymptotic conditional
independence to a factorization of the limiting density, and use it to model
multivariate tails.
