We devise the first rigorous significance test for hyperuniformity with sensitive results, even for a single sample. Our starting point is a detailed study of the empirical Fourier transform of a stationary point process on $\mathbb{R}^d$. For large system sizes, we derive the asymptotic covariances and prove a multivariate central limit theorem (CLT). The scattering intensity is then used as the standard estimator of the structure factor. The above CLT holds for a preferably large class of point processes, and whenever this is the case, the scattering intensity satisfies a multivariate limit theorem as well. Hence, we can use the likelihood ratio principle to test for hyperuniformity. Remarkably, the asymptotic distribution of the resulting test statistic is universal under the null hypothesis of hyperuniformity. We obtain its explicit form from simulations with very high accuracy. The novel test precisely keeps a nominal significance level for hyperuniform models, and it rejects non-hyperuniform examples with high power even in borderline cases. Moreover, it does so given only a single sample with a practically relevant system size.

翻译：我们设计了具有敏感结果的超统一性的第一个严格意义测试, 即使是单一样本。 我们的出发点是详细研究用$\ mathbb{R ⁇ d$对固定点进程进行的经验性Fourier变换。 对于大系统大小, 我们得出无症状的共变量, 并证明其为多变中央限制理论( CLT ) 。 然后, 分散强度被用作结构要素的标准估计器。 上面的 CLT 保存着一大部分的点进程, 并且当出现这种情况时, 散射强度也满足了一个多变数的定点。 因此, 我们可以使用概率比原则来测试超统一性。 值得注意的是, 由此产生的测试统计的无症状分布在超统一性的空假设下是普遍的。 我们从极精确的模拟中获得了清晰的形态。 新的测试精确地保持了超统一模型的名义意义, 并且它拒绝高功率的不统一示例, 即使是在边缘案例中。 此外, 它只给出了一个实际相关的系统大小的单一样本。