Spectra of some self-exciting and mutually exciting point processes

TLDR

Recent work on point process analysis has focused on spectral methods, yet theoretical models remain sparse. This paper investigates the theoretical properties of a class of processes, particularly their point spectrum and covariance density functions. The authors show that a self‑exciting process can share the same second‑order properties as a doubly stochastic process, making them indistinguishable by standard data‑analysis methods.

Abstract

In recent years methods of data analysis for point processes have received some attention, for example, by Cox & Lewis (1966) and Lewis (1964). In particular Bartlett (1963a, b) has introduced methods of analysis based on the point spectrum. Theoretical models are relatively sparse. In this paper the theoretical properties of a class of processes with particular reference to the point spectrum or corresponding covariance density functions are discussed. A particular result is a self-exciting process with the same second-order properties as a certain doubly stochastic process. These are not distinguishable by methods of data analysis based on these properties.