Functional Limits for Hawkes Processes by Child-Event Dispersion

Summary

This paper examines long-run scaling limits for Hawkes processes, with results shaped by the average number and dispersion of child events. For subcritical processes, it establishes functional laws of large numbers and central limit theorems under relatively weak assumptions on the excitation kernel. The resulting limits depend on how dispersed child events are.

At criticality, weakly dispersed offspring do not follow the usual functional central limit behavior: rescaled intensities instead approach a CIR-type process without mean reversion, and rescaled counts approach its integral. The paper gives a Wasserstein-distance bound for the rate of this convergence. With heavily dispersed offspring, critical processes retain functional limit theorems, but exhibit long-range dependence. These are theoretical results about process scaling; the document does not report a trading strategy or empirical market test.

Key ideas

• The average and dispersion of child events determine key long-run properties of Hawkes processes.
• Subcritical processes admit functional laws of large numbers and central limit theorems under minimal kernel conditions.
• Critical processes with weakly dispersed offspring have CIR-type limits instead of standard functional central limit behavior.
• Critical processes with heavily dispersed offspring satisfy functional limit theorems and show long-range dependence.
• A Wasserstein bound quantifies convergence to the weak-dispersion critical limit.

Tags

# Functional Limit Theorems for Hawkes Processes


# Functional Limit Theorems for Hawkes Processes









We prove that the long-run behavior of Hawkes processes is fully determined by the average number and the dispersion of child events. For subcritical processes we provide FLLNs and FCLTs under minimal conditions on the kernel of the process with the precise form of the limit theorems depending strongly on the dispersion of child events. For a critical Hawkes process with weakly dispersed child events, functional central limit theorems do not hold. Instead, we prove that the rescaled intensity processes and rescaled Hawkes processes behave like CIR-processes without mean-reversion, respectively integrated CIR-processes. We provide the rate of convergence by establishing an upper bound on the Wasserstein distance between the distributions of rescaled Hawkes process and the corresponding limit process. By contrast, critical Hawkes process with heavily dispersed child events share many properties of subcritical ones. In particular, functional limit theorems hold. However, unlike subcritical processes critical ones with heavily dispersed child events display long-range dependencies.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.