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

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.