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Cumulants of Hawkes point processes

Stojan Jovanović*

John Hertz†

Stefan Rotter‡

  • Bernstein Center Freiburg & Faculty of Biology, University of Freiburg, 79104 Freiburg im Breisgau, Germany and KTH Royal Institute of Technology, 10691 Stockholm, Sweden

  • Institute for Neuroscience and Pharmacology and Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark and NORDITA, KTH Royal Institute of Technology and Stockholm University, 10691 Stockholm, Sweden

  • Bernstein Center Freiburg & Faculty of Biology, University of Freiburg, 79104 Freiburg im Breisgau, Germany

  • *stojan.jovanovic@bcf.uni-freiburg.de
  • †hertz@nbi.dk
  • ‡stefan.rotter@biologie.uni-freiburg.de
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Phys. Rev. E 91, 042802 – Published 7 April, 2015

DOI: https://doi.org/10.1103/PhysRevE.91.042802

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.

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Phys. Rev. E 91, 042802– Published 7 April, 2015
  • Received 19 September 2014

©2015 American Physical Society

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