  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.
