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

We derive so-called weak and strong \textit{max-laws of large numbers} for $$ for zero mean stochastic triangular arrays $\{x_{i,n,t}$ $:$ $1$ $\leq$ $t$ $\leq n\}_{n\geq 1}$, with dimension counter $i$ $=$ $1,...,k_{n}$ and dimension $$ $\rightarrow$ $\infty$. Rates of convergence are also analyzed based on feasible sequences $\{k_{n}\}$. We work in three dependence settings: independence, Dedecker and Prieur's (2004) $\tau$-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 $k_{n}$ $\rightarrow$ $\infty$; and ($iii$) test a high dimensional parameter after partialling out a fixed dimensional nuisance parameter in a linear time series regression model under $\tau$% -mixing.