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 $n$.
Categories: math.ST stat.TH