$\mathcal{L}_{q}$-maximal inequality for high dimensional means under dependence

TL;DR

This paper establishes an $ ext{L}_q$-maximal inequality for high-dimensional dependent random variables, providing bounds that facilitate Gaussian approximation in complex dependence settings, with applications to statistical inference.

Contribution

It introduces a novel $ ext{L}_q$-maximal inequality for dependent high-dimensional data, extending classical results to dependent structures with applications to Gaussian approximation.

Findings

Bounds depend on $ ext{ln}(p)$ and $ ext{l}_ ext{infty}$ moments.

Gaussian approximation errors tend to zero under certain dependence conditions.

Results applicable to heterogeneous mixing and physical dependence scenarios.

Abstract

We derive an $\mathcal{L}_{q}$-maximal inequality for zero mean dependent random variables $\{x_{t}\}_{t=1}^{n}$ on $\mathbb{R}^{p}$, where $p$ $>>$ $% n $ is allowed. The upper bound is a familiar multiple of $\ln (p)$ and an $% l_{\infty }$ moment, as well as Kolmogorov distances based on Gaussian approximations $(\rho _{n},\tilde{\rho}_{n})$, derived with and without negligible truncation and sub-sample blocking. The latter arise due to a departure from independence and therefore a departure from standard symmetrization arguments. Examples are provided demonstrating $(\rho _{n},% \tilde{\rho}_{n})$ $\rightarrow $ $0$ under heterogeneous mixing and physical dependence conditions, where $(\rho _{n},\tilde{\rho}_{n})$ are multiples of $\ln (p)/n^{b}$ for some $b$ $>$ $0$ that depends on memory, tail decay, the truncation level and block size.