# Stein's method for functions of multivariate normal random variables

Research paper by **Robert E. Gaunt**

Indexed on: **30 Jul '15**Published on: **30 Jul '15**Published in: **Mathematics - Probability**

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#### Abstract

It is a well-known fact that if the random vector $\mathbf{W}$ converges in
distribution to a multivariate normal random variable $\Sigma^{1/2}\mathbf{Z}$,
then $g(\mathbf{W})$ converges in distribution to $g(\Sigma^{1/2}\mathbf{Z})$
if $g$ is continuous. In this paper, we develop a general method for deriving
bounds on the distributional distance between $g(\mathbf{W})$ and
$g(\Sigma^{1/2}\mathbf{Z})$. To illustrate this method, we obtain several
bounds for the case that the $j$-component of $\mathbf{W}$ is given by
$W_j=\frac{1}{\sqrt{n}}\sum_{i=1}^nX_{ij}$, 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 $n^{-p/2}$. We apply these
general bounds to some examples concerning asymptotically chi-square,
variance-gamma and chi distributed statistics.