$\mathcal{L}_{q}$-maximal inequality for high dimensional means under dependence
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909621201928192 |
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| author | Hill, Jonathan B. |
| author_facet | Hill, Jonathan B. |
| contents | We derive an $\mathcal{L}_{q}$-maximal inequality for zero mean dependent random variables $\{x_{t}\}_{t=1}^{n}$ on $\mathbb{R}^{p}$, where $p$ $>>$ $% n $ is allowed. The upper bound is a familiar multiple of $\ln (p)$ and an $% l_{\infty }$ moment, as well as Kolmogorov distances based on Gaussian approximations $(ρ_{n},\tildeρ_{n})$, derived with and without negligible truncation and sub-sample blocking. The latter arise due to a departure from independence and therefore a departure from standard symmetrization arguments. Examples are provided demonstrating $(ρ_{n},% \tildeρ_{n})$ $\rightarrow $ $0$ under heterogeneous mixing and physical dependence conditions, where $(ρ_{n},\tildeρ_{n})$ are multiples of $\ln (p)/n^{b}$ for some $b$ $>$ $0$ that depends on memory, tail decay, the truncation level and block size. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_17800 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $\mathcal{L}_{q}$-maximal inequality for high dimensional means under dependence Hill, Jonathan B. Probability Statistics Theory 60F10, 60-F25 We derive an $\mathcal{L}_{q}$-maximal inequality for zero mean dependent random variables $\{x_{t}\}_{t=1}^{n}$ on $\mathbb{R}^{p}$, where $p$ $>>$ $% n $ is allowed. The upper bound is a familiar multiple of $\ln (p)$ and an $% l_{\infty }$ moment, as well as Kolmogorov distances based on Gaussian approximations $(ρ_{n},\tildeρ_{n})$, derived with and without negligible truncation and sub-sample blocking. The latter arise due to a departure from independence and therefore a departure from standard symmetrization arguments. Examples are provided demonstrating $(ρ_{n},% \tildeρ_{n})$ $\rightarrow $ $0$ under heterogeneous mixing and physical dependence conditions, where $(ρ_{n},\tildeρ_{n})$ are multiples of $\ln (p)/n^{b}$ for some $b$ $>$ $0$ that depends on memory, tail decay, the truncation level and block size. |
| title | $\mathcal{L}_{q}$-maximal inequality for high dimensional means under dependence |
| topic | Probability Statistics Theory 60F10, 60-F25 |
| url | https://arxiv.org/abs/2505.17800 |