  For a multivariate normal distribution, the sparsity of the covariance and
precision matrices encodes complete information about independence and
conditional independence properties. For general distributions, the covariance
and precision matrices reveal correlations and so-called partial correlations
between variables, but these do not, in general, have any correspondence with
respect to independence properties. In this paper, we prove that, for a certain
class of non-Gaussian distributions, these correspondences still hold, exactly
for the covariance and approximately for the precision. The distributions --
sometimes referred to as "nonparanormal" -- are given by diagonal
transformations of multivariate normal random variables. We provide several
analytic and numerical examples illustrating these results.
