  Stein's method is used to obtain two theorems on multivariate normal
approximation. Our main theorem, Theorem 1.2, provides a bound on the distance
to normality for any nonnegative random vector. Theorem 1.2 requires
multivariate size bias coupling, which we discuss in studying the approximation
of distributions of sums of dependent random vectors. In the univariate case,
we briefly illustrate this approach for certain sums of nonlinear functions of
multivariate normal variables. As a second illustration, we show that the
multivariate distribution counting the number of vertices with given degrees in
certain random graphs is asymptotically multivariate normal and obtain a bound
on the rate of convergence. Both examples demonstrate that this approach may be
suitable for situations involving non-local dependence. We also present Theorem
1.4 for sums of vectors having a local type of dependence. We apply this
theorem to obtain a multivariate normal approximation for the distribution of
the random $p$-vector which counts the number of edges in a fixed graph both of
whose vertices have the same given color when each vertex is colored by one of
$p$ colors independently. All normal approximation results presented here do
not require an ordering of the summands related to the dependence structure.
This is in contrast to hypotheses of classical central limit theorems and
examples, which involve e.g., martingale, Markov chain, or various mixing
assumptions.

Probability