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 wihte 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.

中文翻译：

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标记霍克斯点测度的函数极限定理

本文为标记的霍克斯点测度及其相应的散粒噪声过程建立了大数泛函定律和泛函中心极限定理。我们证明，归一化随机度量可以通过高斯白噪声过程加上相关一维布朗运动的适当提升图的总和来近似分布。Brownian 是由事件的自退出到达产生的。我们应用霍克斯点测量的极限定理来分析宿主中萌芽微生物的种群动态。