Star Discrepancy of Double-Infinite Random Matrices

• The paper's main contribution is providing dimension-sensitive, non-asymptotic bounds on star discrepancy using optimal covering numbers and dyadic chaining techniques.
• It employs probabilistic methods with union-bound confidence constraints to derive explicit constants and improved error rates for high-dimensional integration.
• The work underpins practical applications in quasi-Monte Carlo integration and uncertainty quantification by ensuring controlled geometric uniformity in random point sets.

The star discrepancy of double-infinite random matrices quantifies the deviation between empirical measures of high-dimensional random point sets and the uniform distribution over $[0,1]^d$. This metric is central to geometric discrepancy theory and underpins error bounds for quasi-Monte Carlo (QMC) integration and probabilistic sampling in high dimensions. Recent advances have produced sharp, non-asymptotic bounds for such matrices by leveraging optimal covering numbers for axis-parallel boxes along with dyadic chaining techniques, resulting in explicitly computable constants and improved error rates. The subject evolves from classical existence results given by Heinrich et al. to probabilistic constructions via random matrices (Aistleitner), dimension-sensitive refinements (Fiedler, Gnewuch, Weiss), and computationally efficient deterministic analogues using lacunary systems (Löbbe).

1. Definitions and Foundational Concepts

Double-Infinite Random Matrix Model:

A double-infinite random matrix is a family $(X_{n,i})_{n\geq 1, i \geq 1}$ of i.i.d.\ uniform $[0,1]$ random variables. For fixed $d$ and $N$, the first $N$ rows and $d$ columns define the random point set

$$\mathbf{X}_{N,d} = \bigl\{\mathbf{X}_d^{(n)} : n=1,\dots,N\bigr\} \subset [0,1]^d, \quad \mathbf{X}_d^{(n)} = (X_{n,1}, ..., X_{n,d})$$

Star Discrepancy:

The star discrepancy of an $N$-point set $\mathbf{P}_{N,d} \subset [0,1]^d$ is

$$D_N^d(\mathbf{P}_{N,d}) = \sup_{\boldsymbol{\alpha} \in [0,1]^d} \left| \frac{1}{N} \#\{\mathbf{x}_n \in [0, \boldsymbol{\alpha})\} - \prod_{j=1}^d \alpha_j \right|$$

In probabilistic constructions, the notation $D_N^d$ refers to the discrepancy of the above random point set.

Lacunary and Halton Constructions:

Lacunary systems utilize rapidly expanding matrix sequences to mimic independence efficiently. Hybrid schemes combine Halton coordinates and lacunary multipliers to produce deterministic analogues, trading off statistical independence for digit efficiency in simulation (Löbbe, 2014).

2. Main Non-Asymptotic Discrepancy Bounds

Recent results provide non-asymptotic high-probability bounds on the star discrepancy for every $N$-row, $d$-column submatrix of double-infinite random matrices. For $d \geq 3$, one has

$$D_N^d \leq \sqrt{\alpha A_d + \beta B \frac{\ln \log_2 N}{d}\sqrt{\frac{d}{N}}}$$

with universal $B=178$, explicit series for $A_d$ satisfying $A_3 \leq 745$ and $A_2 \leq 915$, and parameters $\alpha, \beta > 1$ determined by the union-bound confidence constraints (Xu et al., 27 Dec 2025).

Dimension-Dependent Constants:

Earlier bounds used fixed $A=1165$; modern results replace this with dimension-sensitive $A_d$, strictly decreasing for $d>1$ and achieving at least $14\%$ improvement for all $d \geq 3$ (Fiedler et al., 2023, Xu et al., 27 Dec 2025).

Confidence Trade-off:

Probability of bound satisfaction is $1 - (\zeta(\alpha)-1)(\zeta(\beta)-1)$, with $\alpha, \beta$ tuned to desired confidence. For $\epsilon=0.05$, typical choices are $\alpha \approx 2.5$, $\beta \approx 2.2$ (Xu et al., 27 Dec 2025).

3. Covering Numbers and Chaining Methods

Optimal Axis-Parallel Covering Numbers:

A finite set $\Gamma \subset [0,1)^d$ is a $\delta$-cover if every $\mathbf{y} \in [0,1)^d$ can be trapped between elements of $\Gamma$ so box volume changes no more than $\delta$. The minimal cardinality $\mathcal{N}_\parallel(d,\delta)$ yields entropy control for chaining.

For $d \geq 3$ and all $\delta$,

$$\mathcal{N}_\parallel(d,\delta) \leq \frac{e^d}{\sqrt{2\pi d}}\delta^{-d}$$

These estimates underpin dyadic chaining analyses and drive improvements in discrepancy constants [(Xu et al., 27 Dec 2025), Gnewuch 2024].

Dyadic Chaining:

Anchored boxes are approximated by a chain of boxes using $2^{-k}$-covers, partitioning them into increments $I_k$ of bounded measure. Union bounds over these increments, controlled via maximal Bernstein inequalities, aggregate small error probabilities across all scales and dimensions.

4. Proof Strategies and Key Techniques

Maximal Bernstein Inequality:

For each increment $I$ at scale $k$, Bernstein's inequality controls deviations:

$$P\left(\max_{1 \leq n \leq 2^{M+1}} \sum_{i=1}^n X_i > t\right) \leq 2 \exp\left(-\frac{t^2}{2\sum\sigma_i^2 + 2Mt/3}\right)$$

$X_i = 1_I(\mathbf{X}_d^{(i)}) - \lambda(I)$, with variance and size constraints inherited from the box measure.

Union Bound Over Scales:

Summation of error probabilities across all increment classes $\mathscr{A}_k$, scales $k$, and dimensions $d$ yields an aggregate failure probability matching the union-bound constraint. Threshold selection for deviations balances entropy versus variance, delivering finite summability over the chaining hierarchy.

Explicit Constant Computation:

Numerical summation of series for $A_d$ (truncated at $k \sim 20$) produces sharp constants. For example, $A_3 \leq 745$ compared to former bounds $A_3 \leq 868$ (Xu et al., 27 Dec 2025, Fiedler et al., 2023).

5. Computational Efficiency and Lacunary Models

Traditional Monte Carlo simulation in $d$ dimensions with $N$ points and precision $H$ uses $dNH$ digits. Lacunary-Halton constructions reduce this to $O(dH + d \log d \, N)$, capitalizing on deterministic multiplier structure and precomputed seeds (Löbbe, 2014). The lacunary regime simulates independence via exponentially growing multipliers, upholding discrepancy control with drastically reduced random-input cost.

6. Applications and Implications

Quasi-Monte Carlo Integration:

Koksma–Hlawka inequality relates star discrepancy to integration error:

$$\left| \frac{1}{N} \sum_{n=1}^N f(\mathbf{X}_d^{(n)}) - \int_{[0,1]^d} f \right| \leq D_N^d \; V(f)$$

Guarantees on $D_N^d$ yield rigorous error and confidence bounds for randomized QMC estimators in high dimensions (Xu et al., 27 Dec 2025, Fiedler et al., 2023).

Uncertainty Quantification:

Non-asymptotic high-probability bounds facilitate confidence bands in Bayesian inference, PDE simulations, and robust numerical analysis where sample reliability in high dimensions is essential.

Discrepancy Theory Benchmarks:

Recent results set new benchmarks for $N \gg d$ regimes, bringing minimax rates up to an explicit $\sqrt{d/N}$, including controlled logarithmic corrections. These constructions represent the sharpest known “worst-case” guarantees for random matrix-induced geometries, motivating further algorithmic research.

Comparative Table: Main Non-Asymptotic Discrepancy Bounds

Paper / Model	Discrepancy Bound Structure	Notable Constant(s)
Heinrich et al. (2001)	$C \sqrt{d/N}$	$C$ unspecified
Aistleitner–Weimar (2013)	$$\sqrt{\alpha} \sqrt{A + B\,\frac{\ln(\log_2 N)}{d}\sqrt{\frac{d}{N}}}$$	$A = 1165, B=178$
Fiedler, Gnewuch, Weiss (2023)	$$\sqrt{\alpha A_d + \beta B\,\frac{\ln(\log_2 N)}{d}\sqrt{\frac{d}{N}}}$$	$A_3 \leq 868$
Gnewuch (2024), present (2025)	$$\sqrt{\alpha A_d + \beta B\,\frac{\ln(\log_2 N)}{d}\sqrt{\frac{d}{N}}}$$	$A_3 \leq 745$
Löbbe (2014), lacunary construction	$(2576 + 357 \log \varepsilon^{-1}) \sqrt{d/N}$	explicit via $\varepsilon$

Significance:

The explicit, dimension-dependent constants and probabilistic bounds established in recent work (Xu et al., 27 Dec 2025, Fiedler et al., 2023) represent the state of the art in controlling the geometric uniformity of high-dimensional random point sets, fundamental for both theoretical and practical advances in numerical computation, uncertainty analysis, and sampling theory.