Cumulants of Hawkes point processes

TL;DR

This paper derives explicit formulas for the cumulant densities of multivariate Hawkes processes, providing a comprehensive mathematical framework for understanding their complex interactions and correlations.

Contribution

It generalizes previous results by providing closed-form expressions for cumulant densities using a Poisson cluster process representation.

Findings

Explicit cumulant density formulas for multivariate Hawkes processes

Method to compute integrated cumulants and average activity measures

Enhanced understanding of complex event interactions

Abstract

We derive explicit, closed-form expressions for the cumulant densities of a multivariate, self-exciting Hawkes point process, generalizing a result of Hawkes in his earlier work on the covariance density and Bartlett spectrum of such processes. To do this, we represent the Hawkes process in terms of a Poisson cluster process and show how the cumulant density formulas can be derived by enumerating all possible "family trees", representing complex interactions between point events. We also consider the problem of computing the integrated cumulants, characterizing the average measure of correlated activity between events of different types, and derive the relevant equations.