We address the challenge of estimating the hyperuniformity exponent $\alpha$ of a spatial point process, given only one realization of it. Assuming that the structure factor $S$ of the point process follows a vanishing power law at the origin (the typical case of a hyperuniform point process), this exponent is defined as the slope near the origin of $\log S$. Our estimator is built upon the (expanding window) asymptotic variance of some wavelet transforms of the point process. By combining several scales and several wavelets, we develop a multi-scale, multi-taper estimator $\widehat{\alpha}$. We analyze its asymptotic behavior, proving its consistency under various settings, and enabling the construction of asymptotic confidence intervals for $\alpha$ when $\alpha < d$ and under Brillinger mixing. This construction is derived from a multivariate central limit theorem where the normalisations are non-standard and vary among the components. We also present a non-asymptotic deviation inequality providing insights into the influence of tapers on the bias-variance trade-off of $\widehat{\alpha}$. Finally, we investigate the performance of $\widehat{\alpha}$ through simulations, and we apply our method to the analysis of hyperuniformity in a real dataset of marine algae.

翻译：我们探讨了在仅给定空间点过程一次实现的情况下，估计其超均匀性指数 $\alpha$ 的挑战。假设该点过程的结构因子 $S$ 在原点处遵循幂律消失规律（超均匀点过程的典型情况），该指数定义为 $\log S$ 在原点附近的对数斜率。我们的估计器建立在点过程某些小波变换的（扩展窗口）渐近方差基础上。通过结合多个尺度和多种小波，我们开发了一种多尺度、多锥度估计器 $\widehat{\alpha}$。我们分析了其渐近行为，证明了其在多种设置下的一致性，并能够在 $\alpha < d$ 和 Brillinger 混合条件下为 $\alpha$ 构建渐近置信区间。该构造源自一个多元中心极限定理，其中各分量的归一化方式是非标准且互异的。我们还提出了一个非渐近偏差不等式，以揭示锥度对 $\widehat{\alpha}$ 偏差-方差权衡的影响。最后，我们通过模拟研究了 $\widehat{\alpha}$ 的性能，并将我们的方法应用于海藻真实数据集的超均匀性分析中。