Fourier analysis of spatial point processes

TL;DR

This paper introduces frequency domain methods using Fourier transforms to analyze the second-order structure of spatial point processes, providing new inferential tools and estimators with proven asymptotic properties.

Contribution

It develops novel frequency domain techniques for spatial point processes, including asymptotic distributions and computationally efficient estimators for model parameters.

Findings

DFTs are asymptotically Gaussian and independent under stationarity.

Derived the asymptotic distribution of spectral density estimators.

Proposed a frequency domain inference method that is robust to model misspecification.

Abstract

In this article, we develop comprehensive frequency domain methods for estimating and inferring the second-order structure of spatial point processes. The main element here is on utilizing the discrete Fourier transform (DFT) of the point pattern and its tapered counterpart. Under second-order stationarity, we show that both the DFTs and the tapered DFTs are asymptotically jointly independent Gaussian even when the DFTs share the same limiting frequencies. Based on these results, we establish an $\alpha$-mixing central limit theorem for a statistic formulated as a quadratic form of the tapered DFT. As applications, we derive the asymptotic distribution of the kernel spectral density estimator and establish a frequency domain inferential method for parametric stationary point processes. For the latter, the resulting model parameter estimator is computationally tractable and yields meaningful interpretations even in the case of model misspecification. We investigate the finite sample performance of our estimator through simulations, considering scenarios of both correctly specified and misspecified models.