  We provide novel characterizations of multivariate normality that incorporate
both the characteristic function and the moment generating function, and we
employ these results to construct a class of affine invariant, consistent and
easy-to-use goodness-of-fit tests for normality. The test statistics are
suitably weighted $L^2$-statistics, and we provide their asymptotic behavior
both for i.i.d. observations as well as in the context of testing that the
innovation distribution of a multivariate GARCH model is Gaussian. We also
study the finite-sample behavior of the new tests and compare the new criteria
with alternative existing tests.
