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

Abstract

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) $\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.