Max-laws of large numbers for weakly dependent high dimensional arrays with applications

We derive so-called weak and strong $\mathit{\text{max-laws}}\mathit{\text{of}}\mathit{\text{large}}\mathit{\text{numbers}}$ for $$ for zero mean stochastic triangular arrays $\{x_{i,n,t}$ $:$ $1$ $\le$ $t$ $\le n{\}}_{n\ge 1}$, with dimension counter $i$ $=$ $1,\mathrm{.}\mathrm{.}\mathrm{.},k_n$ and dimension $$ $\to$ $\mathrm{\infty }$. Rates of convergence are also analyzed based on feasible sequences $\{k_n\}$. We work in three dependence settings: independence, Dedecker and Prieur’s (2004) $\tau$-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 $\alpha$-mixing or physical dependence; ($ii$) extend correlation screening, or marginal regressions, to physical dependent data with diverging dimension $k_n$ $\to$ $\mathrm{\infty }$; and ($iii$) test a high dimensional parameter after partialling out a fixed dimensional nuisance parameter in a linear time series regression model under $\tau$% -mixing.