  The stability of the Ghurye-Olkin (GO) characterization of Gaussian vectors
is analyzed using a partition of the vectors into equivalence classes defined
by their matrix factors. The sum of the vectors in each class is near-Gaussian
in the characteristic function (c.f.) domain if the GO independence condition
is approximately met in the c.f. domain. All vectors have the property that any
vector projection is near-Gaussian in the distribution function (d.f.) domain.
The proofs of these c.f. and d.f. stabilities use tools that establish the
stabilities of theorems by Kac-Bernstein and Cram\'er, respectively. The
results are used to prove stability theorems for differential entropies of
Gaussian vectors and blind source separation of non-Gaussian sources.
