Max-laws of large numbers for weakly dependent high dimensional arrays with applications
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912937822650368 |
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| author | Hill, Jonathan B. |
| author_facet | Hill, Jonathan B. |
| contents | We derive so-called weak and strong \textit{max-laws of large numbers} for $% \max_{1\leq i\leq k_{n}}|1/n\sum_{t=1}^{n}x_{i,n,t}|$ for zero mean stochastic triangular arrays $\{x_{i,n,t}$ $:$ $1$ $\leq $ $t$ $\leq n\}_{n\geq 1}$, with dimension counter $i$ $=$ $1,...,k_{n}$ and dimension $% k_{n}$ $\rightarrow $ $\infty $. Rates of convergence are also analyzed based on feasible sequences $\{k_{n}\}$. We work in three dependence settings: independence, Dedecker and Prieur's (2004) $τ$-mixing and Wu's (2005) physical dependence. We initially ignore cross-coordinate $i$ dependence as a benchmark. We then work with martingale, nearly martingale, and mixing coordinates to deliver improved bounds on $k_{n}$. Finally, we use the results in three applications, each representing a key novelty: we ($i$) bound $k_{n}$\ for a max-correlation statistic for regression residuals under $α$-mixing or physical dependence; ($ii$) extend correlation screening, or marginal regressions, to physical dependent data with diverging dimension $k_{n}$ $\rightarrow $ $\infty $; and ($iii$) test a high dimensional parameter after partialling out a fixed dimensional nuisance parameter in a linear time series regression model under $τ$% -mixing. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_22423 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Max-laws of large numbers for weakly dependent high dimensional arrays with applications Hill, Jonathan B. Statistics Theory 62E99, 60F99, 60F10 We derive so-called weak and strong \textit{max-laws of large numbers} for $% \max_{1\leq i\leq k_{n}}|1/n\sum_{t=1}^{n}x_{i,n,t}|$ for zero mean stochastic triangular arrays $\{x_{i,n,t}$ $:$ $1$ $\leq $ $t$ $\leq n\}_{n\geq 1}$, with dimension counter $i$ $=$ $1,...,k_{n}$ and dimension $% k_{n}$ $\rightarrow $ $\infty $. Rates of convergence are also analyzed based on feasible sequences $\{k_{n}\}$. We work in three dependence settings: independence, Dedecker and Prieur's (2004) $τ$-mixing and Wu's (2005) physical dependence. We initially ignore cross-coordinate $i$ dependence as a benchmark. We then work with martingale, nearly martingale, and mixing coordinates to deliver improved bounds on $k_{n}$. Finally, we use the results in three applications, each representing a key novelty: we ($i$) bound $k_{n}$\ for a max-correlation statistic for regression residuals under $α$-mixing or physical dependence; ($ii$) extend correlation screening, or marginal regressions, to physical dependent data with diverging dimension $k_{n}$ $\rightarrow $ $\infty $; and ($iii$) test a high dimensional parameter after partialling out a fixed dimensional nuisance parameter in a linear time series regression model under $τ$% -mixing. |
| title | Max-laws of large numbers for weakly dependent high dimensional arrays with applications |
| topic | Statistics Theory 62E99, 60F99, 60F10 |
| url | https://arxiv.org/abs/2505.22423 |