Sharp Non-Asymptotic Bounds for the Star Discrepancy of Double-Infinite Random Matrices via Optimal Covering Numbers

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 $\mathbf{\text{optimal}}\mathbf{\text{covering}}\mathbf{\text{numbers}}\mathbf{\text{for}}\mathbf{\text{axis-parallel}}\mathbf{\text{boxes}}$ (Gnewuch, 2024) into the dyadic chaining framework, we achieve $\mathbf{\text{explicitly}}\mathbf{\text{computable}}\mathbf{\text{constants}}$ that improve upon all previously known bounds. For dimension $d\ge 3$, we prove that with high probability, $$D_N^d\le \sqrt{\alpha A_d+\beta B\frac{\ln {\log }_{2}N}{d}}\sqrt{\frac{d}{N}},$$ where $A_d$ is given by an explicit series and satisfies $A_3\le 745$, a $\mathbf{\text{14\%}}\mathbf{\text{improvement}}$ 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 $\mathbf{\text{precise}}\mathbf{\text{trade-off}}$ 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 $\mathbf{\text{quasi-Monte}}\mathbf{\text{Carlo}}\mathbf{\text{integration,}}\mathbf{\text{uncertainty}}\mathbf{\text{quantification,}}\mathbf{\text{and}}\mathbf{\text{high-dimensional}}\mathbf{\text{sampling}}$, and provide a new benchmark for the probabilistic analysis of geometric discrepancy. $\mathbf{\text{Keywords:}}$ Star discrepancy, double-infinite random matrices, covering numbers, dyadic chaining, high-dimensional integration, quasi-Monte Carlo, probabilistic bounds.