Conditional Uniformity and Hawkes Processes

Classic results show that the Hawkes self-exciting point process can be viewed as a collection of temporal clusters, in which exogenously generated initial events give rise to endogenously driven descendant events. This perspective provides distribution cluster’s size through natural connection branching processes, but this is irrespective time. Insight into chronology cluster has been much more elusive. Here, we employ and novel adaptation random time change theorem establish an analog conditional uniformity property enjoyed by Poisson processes. Conditional on number epochs cluster, transformed times are jointly uniform within particular convex polytope. Furthermore, find polytope leads surprising between these continuous state clusters parking functions, discrete objects central enumerative combinatorics closely related Dyck paths lattice. In particular, uniformly functions constitute hidden spines clusters. yields decomposition valuable both methodologically practically, demonstrate application popular Markovian model proposal flexible efficient simulation algorithm.