  We consider the problem of Gaussian approximation for the $\kappa$th
coordinate of a sum of high-dimensional random vectors. Such a problem has been
studied previously for $\kappa=1$ (i.e., maxima). However, in many
applications, a general $\kappa\geq1$ is of great interest, which is addressed
in this paper. We make four contributions: 1) we first show that the
distribution of the $\kappa$th coordinate of a sum of random vectors,
$\boldsymbol{X}= (X_{1},\cdots,X_{p})^{\sf T}= n^{-1/2}\sum_{i=1}^n
\boldsymbol{x}_{i}$, can be approximated by that of Gaussian random vectors and
derive their Kolmogorov's distributional difference bound; 2) we provide the
theoretical justification for estimating the distribution of the $\kappa$th
coordinate of a sum of random vectors using a Gaussian multiplier procedure,
which multiplies the original vectors with i.i.d. standard Gaussian random
variables; 3) we extend the Gaussian approximation result and Gaussian
multiplier bootstrap procedure to a more general case where $\kappa$ diverges;
4) we further consider the Gaussian approximation for a square sum of the first
$d$ largest coordinates of $\boldsymbol{X}$. All these results allow the
dimension $p$ of random vectors to be as large as or much larger than the
sample size $n$.
