What is the Fourier Transform of a Spatial Point Process?

This paper determines how to define a discretely implemented Fourier transform when analysing an observed spatial point process. To develop this transform we answer four questions; first what is the natural definition of a Fourier transform, and what are its spectral moments, second we calculate fourth order moments of the Fourier transform using Campbell's theorem. Third we determine how to implement tapering, an important component for spectral analysis of other stochastic processes. Fourth we answer the question of how to produce an isotropic representation of the Fourier transform of the process. This determines the basic spectral properties of an observed spatial point process.

Automated summary

This paper determines how to define a discretely implemented Fourier transform when analyzing an observed spatial point process. To develop this transform, the authors answer four questions: the natural definition of a Fourier transform and its spectral moments, calculate fourth-order moments of the Fourier transform using Campbell's theorem, determine how to implement tapering for spectral analysis of other stochastic processes, and produce an isotropic representation of the Fourier transform of the process. This determines the basic spectral properties of an observed spatial point process.