On the splitting and aggregating of Hawkes processes

Abstract

We consider the random splitting and aggregating of Hawkes processes. We present the random splitting schemes using the direct approach for counting processes, as well as the immigration–birth branching representations of Hawkes processes. From the second scheme, it is shown that random split Hawkes processes are again Hawkes. We discuss functional central limit theorems (FCLTs) for the scaled split processes from the different schemes. On the other hand, aggregating multivariate Hawkes processes may not necessarily be Hawkes. We identify a necessary and sufficient condition for the aggregated process to be Hawkes. We prove an FCLT for a multivariate Hawkes process under a random splitting and then aggregating scheme (under certain conditions, transforming into a Hawkes process of a different dimension).

Heavy-traffic limits for parallel single-server queues with randomly split Hawkes arrival processes

“…It is shown in [23] that the splitting Hawkes process (N k ) k is a multivariate Hawkes process with conditional intensity…”

“…The corresponding convergence results for Q(n) , Ẑ(n) in Theorem 3.1 will be used. Define [3,23]: Proof. Using the martingales in (6.8), the integral in the lemma can be split into two martingale integrals and a third component with bounded variation, and we show that each term converges to 0 uniformly on [0, T].…”

“…Randomly split Hawkes processes have recently been studied by the present authors [23]. Unlike Poisson processes, the split Hawkes processes become a multivariate Hawkes process with a particular dependence structure (see the covariance function of scaling limits in Proposition 2.1).…”

“…It is therefore also important to first study the model without abandonment. In the meantime, the comparison approach is non-trivial, since this requires us to use the martingale representations for the Hawkes process and the randomly split Hawkes processes [3,23], as well as some martingale inequalities and properties.…”