Expected Star Discrepancy Theory

• Expected star discrepancy theory is the study of the average maximal deviation between empirical point distributions and the uniform measure over anchored, axis-aligned boxes in the unit cube.
• It establishes sharp bounds using i.i.d. sampling with a Θ(√(d/N)) rate and advanced stratified techniques, such as jittered and convex equivolume partitions, achieving improved O(N^(–1/2–1/(2d))) rates.
• The theory underpins practical numerical integration methods, informing quasi-Monte Carlo techniques and optimal partitioning strategies for controlling uniform error in high-dimensional settings.

Expected star discrepancy theory quantifies the typical deviation between the empirical measure of randomly sampled or stratified point sets in the unit cube $[0,1]^d$ and the uniform (Lebesgue) measure, as measured by the supremum over axis-aligned anchored boxes. This field rigorously analyses the average case behavior for major sampling techniques, providing sharp bounds and dissecting partition strategies that minimize expected uniform error in high-dimensional numerical integration, such as quasi-Monte Carlo.

1. Formalism and Classical Results

Let $P_N = \{x_1, \ldots, x_N\} \subset [0,1]^d$. For any $x \in [0,1]^d$, define the anchored box $[0,x) = \prod_{i=1}^d [0, x_i)$ and let $\lambda([0,x)) = \prod_{i=1}^d x_i$ denote its volume. The star discrepancy is

$$D_N^*(P_N) = \sup_{x \in [0,1]^d} \left| \frac{1}{N} \sum_{n=1}^N \mathbf{1}_{[0,x)}(x_n) - \lambda([0,x)) \right| .$$

Expected star discrepancy refers to $\mathbb{E}[D^*_N(P_N)]$, where the average is taken over the randomness inherent in the point set $P_N$, either due to sampling or stratification.

For independent uniform samples (standard Monte Carlo), it is established that

$$\mathbb{E}\left[D_N^*(P_N)\right] = \Theta\left(\sqrt{\frac{d}{N}}\right)$$

with lower and upper bounds matching up to constants, as proven via chaining, VC theory, and concentration inequalities (Doerr, 2012, Gómez et al., 2020, Löbbe, 2014). Aistleitner–Hofer provide explicit tail bounds for i.i.d. samples, and the existence of point sets with discrepancy at most $C\sqrt{d/N}$ is guaranteed by Heinrich–Novak–Wasilkowski–Woźniakowski. However, deterministic constructions matching this remain elusive.

2. Jittered and Stratified Sampling: Theory and Bounds

Jittered sampling partitions $[0,1]^d$ into $m^d = N$ axis-aligned cubes, placing one random point per cube. Pausinger–Steinerberger proved that for constants $c'(d), C'(d) > 0$,

$$c'(d) N^{-1/2-1/(2d)} \leq \mathbb{E}[D_N^*(\text{jitter})] \leq C'(d) N^{-1/2-1/(2d)},$$

strictly outperforming the $\Theta(\sqrt{d/N})$ scaling of i.i.d. random sampling appreciably when $N \gg d^d$ (Pausinger et al., 2015, Doerr, 2021, Xian et al., 2023). Recent refinements improved constants and precise logarithmic corrections, e.g.,

$$\mathbb{E}[D^*_N(\text{jitter})] \leq \frac{d}{m^{d/2+1/2}\left(43.5365\sqrt{\ln\left(\frac{2.2 e m}{d}\right)} + 89.0107 \right)} ,$$

demonstrating tight control over expected error for practical $m, d$ regimes (Xu et al., 25 Dec 2025). Bernstein-type inequalities, slicing, and chaining arguments are standard analytical tools.

3. The Strong Partition Principle

The strong partition principle asserts that for any partition $\{\Omega_i\}$ of $[0,1]^d$ into equal-volume cells, stratification yields strictly smaller expected star discrepancy than Monte Carlo: $$\mathbb{E}[D_N^*(\text{stratified})] < \mathbb{E}[D_N^*(\text{MC})] .$$ Xian–Xu, Xu–Xian, and others generalize this result beyond axis-aligned partitions—any equal-volume measurable partition delivers variance dominance, and thus expectation dominance, over i.i.d. sampling (Xian et al., 2022, Xian et al., 2023, Xu et al., 25 Dec 2025). The proof leverages variance decomposition for each test box and systemic reduction across boundary effects, and is robust to finite $N$ and arbitrary cell geometries. This principle resolves previously open questions in discrepancy theory.

4. Optimal Partitioning: Convex Equivolume Schemes

Advances in partition design have produced explicit classes of convex equivolume partitions (e.g., Stolarsky partitions, $\Omega^*_\sim$) that outperform classical jittered sampling: $$\mathbb{E}[D_N^*(Z)] \leq \mathbb{E}[D_N^*(Y)] < \mathbb{E}[D_N^*(X)],$$ where $Y$ denotes jittered/gridded, $X$ i.i.d., and $Z$ new partition sampling. The optimal expected discrepancy under such tilings obeys

$$\mathbb{E}[D_N^*(Z)] \leq \frac{\sqrt{2d + \frac{2P(\theta)}{3^{d-2}N^{2-1/d}+1}}}{N^{1/2 + 1/(2d)}}$$

with nontrivial improvements in the constant, sharpening estimates even for moderate $d$, $N$ (Xu et al., 29 Dec 2025, Xian et al., 2022). These constructions answer conjectures on the existence of partitions beating jittered sampling in expectation with explicit geometric recipes, and suggest generalizations to even more sophisticated tilings and boundary-shape optimizations.

5. Practical Construction and Applications

Secure pseudorandom bit generators (PRNGs), such as AES–CTR–DRBG, are shown to yield point sets with discrepancy $O(\sqrt{d/N})$ indistinguishable from truly random sets, provided cryptographic security parameter $b$ is high and rounding errors controlled (Gómez et al., 2020). Lacunary point sets, generated by recurrences like $x_{n+1} = (2x_n) \bmod 1$, achieve comparable rates (with a $\sqrt{\log_2 d}$ penalty), but enable fast deterministic generation from a single random seed, advantageous in hardware or reproducibility-critical contexts (Löbbe, 2014).

Quasi-Monte Carlo integration benefits directly from stratification: lower expected star discrepancy translates into smaller uniform error for indicator test functions, justifying widespread adoption of stratified and randomized-QMC schemes in high-dimensional numerical integration.

6. Lower Bounds, Tightness, and Open Problems

Doerr (Doerr, 2012) establishes the matching lower bound for expected star discrepancy of random i.i.d. samples: $\mathbb{E}[D_N^*(P_N)] \geq K \sqrt{\frac{d}{N}},$ for some absolute $K>0$, confirming the $\sqrt{d/N}$ law is tight. For stratified/jittered constructions, the $N^{-1/2-1/(2d)}$ scaling and the role of partition geometry in lowering constants is provably sharp. Open problems concern explicit deterministic constructions matching best random bounds, optimization of leading constants for new partition schemes, and extension of the strong partition principle to alternative notions of discrepancy (e.g., weighted or directional variants) (Xian et al., 2022, Xu et al., 25 Dec 2025, Xu et al., 29 Dec 2025).

7. Summary Table: Expected Star Discrepancy Rates

Sampling Method	Expected Rate $\mathbb{E}[D_N^*]$	Partition Principle Validity
Simple random (Monte Carlo, i.i.d.)	$\Theta(\sqrt{d/N})$	Dominated by stratified schemes
Jittered sampling (axis-aligned cubes)	$O(N^{-1/2-1/(2d)})$, constants improved recently	Strictly better than MC
Convex equivolume partitions	$O(N^{-1/2-1/(2d)})$, strictly smaller constant	Strong partition principle holds
Secure PRNG-based sets (high-security)	$O(\sqrt{d/N})$ (almost indistinguishable from random)	Follows MC behavior
Lacunary sequences (e.g., $x_{n+1}=2x_n$ mod 1)	$O(\sqrt{d \log_2 d/N})$	Slight penalty over MC

These rates define the frontier of expected star discrepancy theory, which now centers on realizing sharp constants, advanced partitioning, and extending the partition principle to broader families of sampling schemes and discrepancy types. Stratification—not just axis-aligned but convex or tailored—is universally preferable to pure random sampling for minimizing expected uniformity error in finite dimensions.