arXiv subjects

Gabriel Mastrilli

Publications and source records attributed to Gabriel Mastrilli.

4 recordsLinked to original sources

arXiv2025

Minimax estimation of the structure factor of spatial point processes

We investigate the problem of estimating the structure factor, or spectra, of stationary spatial point processes. In the first part, we establish a minimax lower bound for this estimation problem, using an approach tailored to second-order properties of spatial point processes. Although not the main focus, this methodology also extends naturally to a minimax lower bound for the estimation of the pair correlation function of spatial point processes. In the second part, we construct a multitaper estimator that achieves the optimal rate of convergence in squared risk. Under a Brillinger-mixing condition, we further establish a chi-square-type concentration bound. Finally, we propose a data-driven procedure for selecting the number of tapers, supported by an oracle inequality, and we demonstrate the practical effectiveness of the method through numerical experiments.

math.ST↗

arXiv2025

Asymptotic fluctuations of smooth linear statistics of independently perturbed lattices

We consider the hyperuniform model of d-dimensional integer lattice perturbed by independent random variables and we investigate the large scale asymptotic fluctuations of smoothed versions of the usual counting statistics, specifically of linear statistics associated to a smooth function with rapid decay at infinity. We highlight three distinct classes of limit, depending on the dimension d and on the tails of the perturbations. On the one hand, we establish that for dimensions larger than two, central limit theorems hold under mild assumptions on the perturbations. This confirms numerical observations from physics, suggesting that even for highly correlated hyperuniform models, large dimensions favor asymptotic normality. On the other hand, in dimension one, the limiting distribution can be Gaussian, non-Gaussian with finite moments of all orders, or stable with parameter strictly between one and two. These two latter results represent rare examples of non-Gaussian limits for smooth linear statistics of hyperuniform point processes of Classes I and II.

math.PR↗

arXiv2024

Estimating the hyperuniformity exponent of point processes

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.

math.ST↗

arXiv2023

Repelled point processes with application to numerical integration

We look at Monte Carlo numerical integration from a stochastic geometry point of view. While crude Monte Carlo estimators relate to linear statistics of a homogeneous Poisson point process (PPP), linear statistics of more regularly spread point processes can yield unbiased estimators with faster-decaying variance, and thus lower integration error. Following this intuition, we introduce a Coulomb repulsion operator, which reduces clustering by slightly pushing the points of a configuration away from each other. Our empirical findings show that applying the repulsion operator to a PPP as well as, intriguingly, to more regular point processes, preserves unbiasedness while reducing the variance of the corresponding Monte Carlo estimator, thus enhancing the method. We prove this variance reduction when the initial point process is a PPP. On the computational side, the complexity of the operator is quadratic and the corresponding algorithm can be parallelized without communication across tasks.

stat.ME↗