{"ID":2824805,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.22557","arxiv_id":"2512.22557","title":"Sharp Non-Asymptotic Bounds for the Star Discrepancy of Double-Infinite Random Matrices via Optimal Covering Numbers","abstract":"We establish sharp non-asymptotic probabilistic bounds for the star discrepancy of double-infinite random matrices -- a canonical model for sequences of random point sets in high dimensions. By integrating the recently proved \\textbf{optimal covering numbers for axis-parallel boxes} (Gnewuch, 2024) into the dyadic chaining framework, we achieve \\textbf{explicitly computable constants} that improve upon all previously known bounds. For dimension $d \\ge 3$, we prove that with high probability, \\[ D_N^d \\le \\sqrt{αA_d + βB \\frac{\\ln \\log_2 N}{d}} \\sqrt{\\frac{d}{N}}, \\] where $A_d$ is given by an explicit series and satisfies $A_3 \\le 745$, a \\textbf{14\\% improvement} over the previous best constant of 868 (Fiedler et al., 2023). For $d=2$, we obtain the currently smallest known constant $A_2 \\le 915$. Our analysis reveals a \\textbf{precise trade-off} between the dimensional dependence and the logarithmic factor in $N$, highlighting how optimal covering estimates directly translate to tighter discrepancy bounds. These results immediately yield improved error guarantees for \\textbf{quasi-Monte Carlo integration, uncertainty quantification, and high-dimensional sampling}, and provide a new benchmark for the probabilistic analysis of geometric discrepancy. \\textbf{Keywords:} Star discrepancy, double-infinite random matrices, covering numbers, dyadic chaining, high-dimensional integration, quasi-Monte Carlo, probabilistic bounds.","short_abstract":"We establish sharp non-asymptotic probabilistic bounds for the star discrepancy of double-infinite random matrices -- a canonical model for sequences of random point sets in high dimensions. By integrating the recently proved \\textbf{optimal covering numbers for axis-parallel boxes} (Gnewuch, 2024) into the dyadic chai...","url_abs":"https://arxiv.org/abs/2512.22557","url_pdf":"https://arxiv.org/pdf/2512.22557v2","authors":"[\"Xiaoda Xu\",\"Jun Xian\"]","published":"2025-12-27T11:09:59Z","proceeding":"math.ST","tasks":"[\"math.ST\"]","methods":"[]","has_code":false}