Cumulants of Hawkes point processes

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.

Figure 1

Evolution of a Poisson cluster process on a network with 4 nodes and 6 directed links. Left column: Hatched nodes are active, and dashed links are transmitting a signal to all their respective neighbors. Right column: A cluster, generated by the arrival of an immigrant from node $A$, evolves sequentially as new offspring are produced. Row 1: A type $A$ immigrant arrives, providing the seed from which a new cluster will emerge. Row 2: The type $A$ immigrant generates 1st generation offspring of types $B$ and $D$, respectively. Together, these constitute the 1st generation of events. Row 3: The 1st generation, type $D$, event generates a single offspring event of type $C$. The event of type $B$ creates two offspring of types $A$ and $C$, respectively. Row 4: Finally, the 2nd generation, type $A$, event is the only one to generate 3 offspring events, of types $B,C$, and $D$, respectively. Bottom: $A$ time sequence of events generated in this 4-generation-long evolution, here displayed by means of a “raster plot,” which indicates the events generated by each node on the time axis.

Figure 2

A schematic representation of all possible family trees, containing events $t_1$ and $t_2$. Two concrete examples of family trees, mapping to the same general schema, are given in the second row. The root, $x$, denotes the original immigrant and $u$ the branching point in the family tree of immigrant $x$, leading to the appearance of offspring $t_1$ and $t_2$. Note that each link connecting two nodes can be, in theory, any number of generations long.

Figure 3

A schematic representation of all possible family trees, containing events $t_1,t_2$, and $t_3$. Once again, the root $x$ denotes the original immigrant, nodes $u$ and $v$ represent the branching points in the family tree of immigrant $x$, and each link between two nodes can be any number of generations long.