$\mathcal{L}_{q}$-maximal inequality for high dimensional means under dependence
Jonathan B. Hill
TL;DR
This work develops an $L_q$-maximal inequality for zero-mean, high-dimensional dependent means with $p$ growing faster than $n$, allowing $p\gg n$ under broad dependence and tail assumptions. The authors bypass the lack of a symmetrization argument by using a blocking/ multiplier bootstrap framework together with a negligibly truncated approximation and Gaussian comparison, yielding Kolmogorov-distance control via $\rho_n$ and $\rho_n^*$. They derive Nemirovski-type bounds in the bounded case and extend to unbounded settings with careful truncation and tail-concentration bounds, obtaining explicit rates under geometric mixing and physical dependence. The results enable high-dimensional inference and LLN-type conclusions for dependent data, with practical guidance on how tail behavior, memory, and block design affect allowable growth of $p$.
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 $(ρ_{n},\tildeρ_{n})$, 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 $(ρ_{n},% \tildeρ_{n})$ $\rightarrow $ $0$ under heterogeneous mixing and physical dependence conditions, where $(ρ_{n},\tildeρ_{n})$ are multiples of $\ln (p)/n^{b}$ for some $b$ $>$ $0$ that depends on memory, tail decay, the truncation level and block size.