  One of the most widely used properties of the multivariate Gaussian
distribution, besides its tail behavior, is the fact that conditional means are
linear and that conditional variances are constant. We here show that this
property is also shared, in an approximate sense, by a large class of
non-Gaussian distributions. We allow for several conditioning variables and we
provide explicit non-asymptotic results, whereby we extend earlier findings of
Hall and Li (1993) and Leeb (2013).
