Symmetrization for high dimensional dependent random variables

Abstract

We establish a generic symmetrization property for dependent random variables $\{x_{t}\}_{t=1}^{n}$ on $\mathbb{R}^{p}$, where $p$ $>>$ $n$ is allowed. We link $\mathbb{E}\psi (\max_{1\leq i\leq p}|1/n\sum_{t=1}^{n}(x_{i,t}$ $-$ $\mathbb{E}x_{i,t})|)$ to $\mathbb{E}\psi (\max_{1\leq i\leq p}|1/n$ $\sum_{t=1}^{n}\eta _{t}(x_{i,t}$ $-$ $\mathbb{E}% x_{i,t})|)$ for non-decreasing convex $\psi $ $:$ $[0,\infty )$ $\rightarrow $ $\mathbb{R}$, where $\{\eta _{t}\}_{t=1}^{n}$ are block-wise independent random variables, with a remainder term based on high dimensional Gaussian approximations that need not hold at a high level. Conventional usage of $% \eta _{t}(x_{i,t}$ $-$ $\tilde{x}_{i,t})$ with $\{\tilde{x}% _{i,t}\}_{t=1}^{n} $ an independent copy of $\{x_{i,t}\}_{t=1}^{n}$, and Rademacher $\eta _{t}$, is not required in a generic environment, although we may trivially replace $\mathbb{E}x_{i,t}$ with $\tilde{x}_{i,t}$. In the latter case with Rademacher $\eta _{t}$ our result reduces to classic symmetrization under independence. We bound and therefore verify the Gaussian approximations in mixing and physical dependence settings, thus bounding $\mathbb{E}\psi (\max_{1\leq i\leq p}|1/n\sum_{t=1}^{n}(x_{i,t}$ $-$ $\mathbb{E}x_{i,t})|)$; and apply the main result to a generic % Nemirovski (2000)-like $\mathcal{L}_{q}$-maximal moment bound for $\mathbb{E}\max_{1\leq i\leq p}|1/n\sum_{t=1}^{n}(x_{i,t}$ $-$ $\mathbb{E}x_{i,t})|^{q}$, $q$ $\geq $ $1$.