Symmetrization for high dimensional dependent random variables

We establish a generic symmetrization property for dependent random variables {xt}t=1n on Rp, where p≫n is allowed. We link Eψ(max1≤i≤p|1/n∑t=1n(xi,t−Exi,t)|) to Eψ(max1≤i≤p|1/n∑t=1nηt(xi,t−Exi,t)|) for non-decreasing convex ψ:[0,∞)→R, where {ηt}t=1n are block-wise independent random variables, with a remainder term based on high dimensional Gaussian approximations that need not hold at a high level (the result holds generally, without imposed dependence and stationarity properties) . Conventional usage of ηt(xi,t−x̃i,t) with {x̃i,t}t=1n an independent copy of {xi,t}t=1n , and Rademacher ηt, is not required in a generic environment, although we may trivially replace Exi,t with x̃i,t . In the latter case with Rademacher ηt our result reduces to classic symmetrization under independence. We bound and therefore verify the Gaussian approximations in mixing and physical dependence settings, thus bounding Eψ(max1≤i≤p|1/n∑t=1n(xi,t−Exi,t)|). We apply the main result to a generic Lq-maximal moment bound for Emax1≤i≤p|1/n∑t=1n(xi,t−Exi,t)|q, q≥ 1.