Conditional Uniformity and Hawkes Processes

TL;DR

This paper introduces a novel conditional uniformity property for Hawkes processes, linking cluster times to combinatorial parking functions, and provides new methods for analysis and simulation of these processes.

Contribution

It establishes a conditional uniformity property for Hawkes process clusters and uncovers a surprising connection to parking functions, enhancing analysis and simulation techniques.

Findings

Conditional times are uniformly distributed within a convex polytope.

Hawkes clusters are connected to parking functions, revealing hidden combinatorial structures.

The new methods improve simulation efficiency for Hawkes processes.

Abstract

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