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. Furthermore, we extend our proposed DFT-based frequency domain methods to a class of non-stationary spatial point processes.

翻译：本文为估计和推断空间点过程的二阶结构，发展了一套全面的频域方法。其核心在于利用点模式的离散傅里叶变换（DFT）及其加窗版本。在二阶平稳性假设下，我们证明了即使DFT共享相同的极限频率，DFT与加窗DFT均渐近地服从联合独立高斯分布。基于这些结果，我们为以加窗DFT的二次型形式构造的统计量建立了一个$\alpha$混合中心极限定理。作为应用，我们推导了核谱密度估计量的渐近分布，并为参数化平稳点过程建立了一种频域推断方法。对于后者，所得模型参数估计量在计算上是易于处理的，即使在模型设定错误的情况下也能产生有意义的解释。我们通过模拟研究，在模型正确设定和错误设定两种情形下，考察了估计量的有限样本表现。此外，我们将所提出的基于DFT的频域方法推广到了一类非平稳空间点过程。