2025/05/31 by Jonathan B. Hill, Hill, Jonathan B.
Mathematics · #60-F10 #60-F25 #FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2506.00547
openalex publication_date 2025/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 - 𝔼% xi,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. Conventional usage of % ηt(xi,t - xi,t) with \x% i,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 xi,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)|); and apply the main result to a generic % Nemirovski (2000)-like Lq-maximal moment bound for 𝔼max1≤ i≤ p|1/n∑t=1n(xi,t - 𝔼xi,t)|q, q ≥ 1.