Estimating the hyperuniformity exponent of spatial point processes

We address the challenge of estimating the hyperuniformity exponent α 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 logS. Our estimator is built upon the (expanding window) asymptotic variance of some shot-noise wavelet transforms, of the point process. By combining several scales and several wavelets, we develop a multi-scale, multi-taper estimator αˆ. We analyze its asymptotic behavior, proving its consistency under various settings, and enabling the construction of asymptotic confidence intervals for α when α<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 αˆ. Finally, we investigate the performance of αˆ through simulations, and we apply our method to the analysis of hyperuniformity in a real dataset of marine algae.