Cumulants of Hawkes point processes

Area of Science:

• Point process theory
• Stochastic modeling
• Time series analysis

Background:

• Hawkes processes model self-exciting phenomena where past events influence future ones.
• Existing work by Hawkes focused on covariance density and Bartlett spectrum.
• Understanding higher-order statistics is crucial for complex systems.

Purpose of the Study:

• Derive explicit, closed-form expressions for cumulant densities of multivariate Hawkes processes.
• Generalize previous findings on covariance and Bartlett spectrum.
• Develop methods for computing integrated cumulants to measure correlated activity.

Main Methods:

• Representing the Hawkes process as a Poisson cluster process.
• Enumerating "family trees" to model event interactions.
• Deriving equations for integrated cumulants.

Main Results:

• Obtained explicit, closed-form expressions for cumulant densities.
• Established a novel connection between Hawkes processes and Poisson cluster processes.
• Derived equations for computing integrated cumulants.

Conclusions:

• The derived formulas offer a comprehensive tool for analyzing multivariate Hawkes processes.
• This work advances the theoretical understanding of self-exciting point processes.
• The methods provide a way to quantify correlated activity between different event types.