ℒ q -maximal inequality for high dimensional means under dependence

We derive an Lq-maximal inequality for zero mean dependent random variables {xt}t=1n on Rp, where p>>n is allowed. The upper bound is a familiar multiple of ln⁡(p) and an l∞ moment, as well as Kolmogorov distances based on Gaussian approximations (ρn,ρ~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 symmetrisation arguments. Examples are provided demonstrating (ρn,ρ~n)→0 under heterogeneous mixing and physical dependence conditions, where (ρn,ρ~n) are multiples of ln⁡(p)/nb for some b>0 that depends on memory, tail decay, the truncation level and block size.