${\mathcal{L}}_q$-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 $p$ $>>$ $$ is allowed. The upper bound is a familiar multiple of $\ln (p)$ and an $$ 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 $({\rho }_n,$ $\to$ $0$ under heterogeneous mixing and physical dependence conditions, where $({\rho }_n,{\tilde{\rho }}_n)$ are multiples of $\ln (p)\mathrm{/}{n}^b$ for some $b$ $>$ $0$ that depends on memory, tail decay, the truncation level and block size.