Max-laws of large numbers for weakly dependent high dimensional arrays with applications
Jonathan B. Hill
TL;DR
The paper develops max-laws of large numbers for the maximum of high-dimensional coordinate-wise means in zero-mean triangular arrays under weak dependence. It formulates and proves max-WLLN and max-SLLN across three dependence regimes—$\tau$-mixing, $\mathcal{L}_p$-physical dependence, and independence—and shows how cross-coordinate dependence can improve allowable growth of the dimension $k_n$. The results are specialized to three econometric applications: a residual max-correlation test, marginal screening with diverging covariates, and a high-dimensional parameter test after partialling out nuisance terms, each accompanied by Gaussian (or non-Gaussian) HD approximations and explicit rate conditions. These contributions provide practical guidelines for HD inference in dependent data and offer a foundation for further extensions to NED and spatial settings. Overall, the work advances HD LLN theory under dependence and demonstrates its relevance for robust high-dimensional testing and screening in time series contexts.
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) $τ$-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.