Cumulants of Hawkes point processes

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.

11 pages, 4 figures

References in corpus (3)

• Branching ratio approximation for the self-exciting Hawkes process
• Hot-spot model for accretion disc variability as random process
• Bursting transition in a linear self-exciting point process

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