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 $\mathbb{E}\ddot{I}ˆ({\max }_{1\le i\le p}|1/n{\sum }_{t=1}^n(x_{i,t}$ $-$ $\mathbb{E}{x}_{i,t})|)$ to $\mathbb{E}\ddot{I}ˆ({\max }_{1\le i\le p}|1/n$ ${\sum }_{t=1}^n\hat{I}{\cdot }_t(x_{i,t}$ $-$ $\mathbb{E}$ for non-decreasing convex $\ddot{I}ˆ$ $:$ $[0,\infty )$ $\to$ $\mathbb{R}$, where $\{\hat{I}{\cdot }_t{\}}_{t=1}^n$ 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 $-$ ${\tilde{x}}_{i,t})$ with $\{\tilde{x}$ an independent copy of $\{x_{i,t}{\}}_{t=1}^n$, and Rademacher $\hat{I}{\cdot }_t$, 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 $\hat{I}{\cdot }_t$ our result reduces to classic symmetrization under independence. We bound and therefore verify the Gaussian approximations in mixing and physical dependence settings, thus bounding $\mathbb{E}\ddot{I}ˆ({\max }_{1\le i\le p}|1/n{\sum }_{t=1}^n(x_{i,t}$ $-$ $\mathbb{E}{x}_{i,t})|)$; and apply the main result to a generic % Nemirovski (2000)-like ${\mathcal{L}}_q$-maximal moment bound for $\mathbb{E}{\max }_{1\le i\le p}|1/n{\sum }_{t=1}^n(x_{i,t}$ $-$ $\mathbb{E}{x}_{i,t}){|}^q$, $q$ $\ge$ $1$.