Estimating the hyperuniformity exponent of point processes

TL;DR

This paper introduces a multi-scale, multi-taper estimator for accurately determining the hyperuniformity exponent of a point process from a single realization, with proven consistency and confidence interval construction.

Contribution

It develops a novel estimator for the hyperuniformity exponent based on wavelet transforms, analyzing its asymptotic properties and demonstrating its effectiveness through simulations and real data application.

Findings

Estimator is consistent under various settings.

Constructs asymptotic confidence intervals for the exponent.

Performs well in simulations and real marine algae data.

Abstract

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.