# Spectral density of point processes

The paper titled,”Spectra of some self-exciting and mutually exciting point processes”, is one of the most widely cited papers in marked point process literature. I guess this was the first paper that explored the complete covariance density function of point processes, and in particular, self exciting and mutually exciting processes. In the time series literature, the covariance of a stationary process at various lags have special meaning. If you consider the generating function of the covariance at various lags and evaluate at a specific complex exponential, you arrive at population spectrum of the time series. The population spectrum thus obtained has several interesting applications.

In this paper, the author starts off with deriving the spectral density of a general point process and then applies this generic form to univariate self-exciting process, multivariate self-exciting and mutually exciting processes. In each of the cases explored, the author writes an equation for covariance density that looks like a renewal equation. By applying Laplace transforms on the obtained equation, the author obtains Laplace transform of the covariance density in terms of the Laplace transform of the propagator function and baseline rate function. For simple exponentials and univariate processes, one can apply inverse Laplace transform to obtain the covariance density. For multivariate process, the math is tedious but the procedure is straightforward. In each of the cases explored, once the covariance density is obtained, it is straightforward to obtain the spectral density of the process.