  We show that a necessary and sufficient condition for the sum of iid random
vectors to converge (under appropriate shifting and scaling) to a multivariate
Gaussian distribution is that the truncated second moment matrix is slowly
varying at infinity. This is more natural than the standard conditions, and
allows for the possibility that the limiting Gaussian distribution is
degenerate (so long as it is not concentrated at a point). We also give
necessary and sufficient conditions for a d-dimensional Levy process to
converge (under appropriate shifting and scaling) to a multivariate Gaussian
distribution as time approaches zero or infinity.
