Dual Induction CLT for High-dimensional m-dependent Data
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866911128321261568 |
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| author | Bong, Heejong Kuchibhotla, Arun Kumar Rinaldo, Alessandro |
| author_facet | Bong, Heejong Kuchibhotla, Arun Kumar Rinaldo, Alessandro |
| contents | We derive novel and sharp high-dimensional Berry--Esseen bounds for the sum of $m$-dependent random vectors over the class of hyper-rectangles exhibiting only a poly-logarithmic dependence in the dimension. Our results hold under minimal assumptions, such as non-degenerate covariances and finite third moments, and exhibit an optimal sample complexity of order $m^{(q-1)/(q-2)}/\sqrt{n}$. Aside from logarithmic terms, the resulting rates match the optimal rates established in the univariate case. When specialized to the sums of independent non-degenerate random vectors, our results produce sharp and, in some cases, optimal rates under the weakest possible conditions. We develop a novel inductive relationship between anti-concentration inequalities and Berry--Esseen bounds inspired by the classical Lindeberg swapping method and the concentration inequality approach for dependent data that may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_14299 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Dual Induction CLT for High-dimensional m-dependent Data Bong, Heejong Kuchibhotla, Arun Kumar Rinaldo, Alessandro Probability Statistics Theory 60B12, 60F05 We derive novel and sharp high-dimensional Berry--Esseen bounds for the sum of $m$-dependent random vectors over the class of hyper-rectangles exhibiting only a poly-logarithmic dependence in the dimension. Our results hold under minimal assumptions, such as non-degenerate covariances and finite third moments, and exhibit an optimal sample complexity of order $m^{(q-1)/(q-2)}/\sqrt{n}$. Aside from logarithmic terms, the resulting rates match the optimal rates established in the univariate case. When specialized to the sums of independent non-degenerate random vectors, our results produce sharp and, in some cases, optimal rates under the weakest possible conditions. We develop a novel inductive relationship between anti-concentration inequalities and Berry--Esseen bounds inspired by the classical Lindeberg swapping method and the concentration inequality approach for dependent data that may be of independent interest. |
| title | Dual Induction CLT for High-dimensional m-dependent Data |
| topic | Probability Statistics Theory 60B12, 60F05 |
| url | https://arxiv.org/abs/2306.14299 |