  By the continuous mapping theorem, if a sequence of $d$-dimensional random
vectors $(\mathbf{W}_n)_{n\geq1}$ converges in distribution to a multivariate
normal random variable $\Sigma^{1/2}\mathbf{Z}$, then the sequence of random
variables $(g(\mathbf{W}_n))_{n\geq1}$ converges in distribution to
$g(\Sigma^{1/2}\mathbf{Z})$ if $g:\mathbb{R}^d\rightarrow\mathbb{R}$ is
continuous. In this paper, we develop Stein's method for the problem of
deriving explicit bounds on the distance between $g(\mathbf{W}_n)$ and
$g(\Sigma^{1/2}\mathbf{Z})$ with respect to smooth probability metrics. We
obtain several bounds for the case that the $j$-component of $\mathbf{W}_n$ is
given by $W_{n,j}=\frac{1}{\sqrt{n}}\sum_{i=1}^nX_{ij}$, where the $X_{ij}$ are
independent. In particular, provided $g$ satisfies certain differentiability
and growth rate conditions, we obtain an order $n^{-(p-1)/2}$ bound, for smooth
test functions, if the first $p$ moments of the $X_{ij}$ agree with those of
the normal distribution. If $p$ is an even integer and $g$ is an even function,
this convergence rate can be improved further to order $n^{-p/2}$. These
convergence rates are shown to be of optimal order. We apply our general bounds
to some examples, which include the distributional approximation of
asymptotically chi-square distributed statistics; the approximation of
expectations of smooth functions of binomial and Poisson random variables;
rates of convergence in the delta method; and a quantitative variance-gamma
approximation of the $D_2^*$ statistic for alignment-free sequence comparison
in the case of binary sequences.
