Functional Limit Theorems for Marked Hawkes Point Measures

TL;DR

This paper develops limit theorems for marked Hawkes point measures, showing their convergence to Gaussian processes, and applies these results to model microbial population dynamics.

Contribution

It introduces new functional limit theorems for marked Hawkes processes and their shot noise processes, advancing the theoretical understanding of self-exciting point measures.

Findings

Normalized measures approximate Gaussian white noise plus a Brownian motion component

Limit theorems enable analysis of complex stochastic systems

Application to microbial population dynamics demonstrates practical relevance

Abstract

This paper establishes a functional law of large numbers and a functional central limit theorem for marked Hawkes point measures and their corresponding shot noise processes. We prove that the normalized random measure can be approximated in distribution by the sum of a Gaussian white noise process plus an appropriate lifting map of a correlated one-dimensional Brownian motion. The Brownian results from the self-exiting arrivals of events. We apply our limit theorems for Hawkes point measures to analyze the population dynamics of budding microbes in a host.