Functional central limit theorems for stationary Hawkes processes and application to infinite-server queues

TL;DR

This paper establishes a functional central limit theorem for stationary Hawkes processes with high baseline intensity, approximating queue length processes in infinite-server queues by Gaussian processes, and extends results to multivariate cases.

Contribution

It introduces a novel functional CLT for stationary Hawkes processes in a high-intensity regime and applies it to analyze infinite-server queues with Hawkes traffic, including multivariate extensions.

Findings

Queue length process approximated by Gaussian process

Explicit covariance function derived for the Gaussian approximation

Limit theorems established for multivariate Hawkes processes

Abstract

A univariate Hawkes process is a simple point process that is self-exciting and has clustering effect. The intensity of this point process is given by the sum of a baseline intensity and another term that depends on the entire past history of the point process. Hawkes process has wide applications in finance, neuroscience, social networks, criminology, seismology, and many other fields. In this paper, we prove a functional central limit theorem for stationary Hawkes processes in the asymptotic regime where the baseline intensity is large. The limit is a non-Markovian Gaussian process with dependent increments. We use the resulting approximation to study an infinite-server queue with high-volume Hawkes traffic. We show that the queue length process can be approximated by a Gaussian process, for which we compute explicitly the covariance function and the steady-state distribution. We also extend our results to multivariate stationary Hawkes processes and establish limit theorems for infinite-server queues with multivariate Hawkes traffic.