Gaussian Approximations for the $k$th coordinate of sums of random vectors
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| Format: | Preprint |
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2024
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| author | Ding, Yixi Li, Qizhai Shi, Yuke Zhang, Wei |
| author_facet | Ding, Yixi Li, Qizhai Shi, Yuke Zhang, Wei |
| contents | We consider the problem of Gaussian approximation for the $κ$th coordinate of a sum of high-dimensional random vectors. Such a problem has been studied previously for $κ=1$ (i.e., maxima). However, in many applications, a general $κ\geq1$ is of great interest, which is addressed in this paper. We make four contributions: 1) we first show that the distribution of the $κ$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 $κ$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 $κ$ 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 $n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_03039 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gaussian Approximations for the $k$th coordinate of sums of random vectors Ding, Yixi Li, Qizhai Shi, Yuke Zhang, Wei Statistics Theory We consider the problem of Gaussian approximation for the $κ$th coordinate of a sum of high-dimensional random vectors. Such a problem has been studied previously for $κ=1$ (i.e., maxima). However, in many applications, a general $κ\geq1$ is of great interest, which is addressed in this paper. We make four contributions: 1) we first show that the distribution of the $κ$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 $κ$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 $κ$ 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 $n$. |
| title | Gaussian Approximations for the $k$th coordinate of sums of random vectors |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2408.03039 |