Hawkes和INAR（$\infty$）过程
Hawkes and INAR($\infty$) processes
摘要：
本文讨论了整数值自回归时间序列（INAR）、Hawkes点过程及其相互关系。除了提出结构类比，我们推导出一个收敛定理。更具体地说，我们推广了著名的INAR（$p$），$p\in\mathbb{N}$，时间序列模型到相应的模型的无限阶：INAR（$\infty$）模型。我们建立的存在性，唯一性，有限的时刻，并给出公式的自协方差函数以及联合矩母函数。此外，我们推导出一个AR（$\infty$），MA（$\infty$），和分支过程表示的模型。我们比较了Hawkes过程与它们的INAR（$\infty$）对应物的性质。给定一个Hawkes过程$N$，在本文的主要定理中，我们构造了一个基于INAR（$\infty$）的点过程族，并证明了它收敛于$N$ . INAR模型和Hawkes模型之间的这种联系在应用中是相关的。
英文摘要：
In this paper, we discuss integer-valued autoregressive time series (INAR), Hawkes point processes, and their interrelationship. Besides presenting structural analogies, we derive a convergence theorem. More specifically, we generalize the well-known INAR($p$), $p\in\mathbb{N}$, time series model to a corresponding model of infinite order: the INAR($\infty$) model. We establish existence, uniqueness, finiteness of moments, and give formulas for the autocovariance function as well as for the joint moment-generating function. Furthermore, we derive an AR($\infty$), an MA($\infty$), and a branching-process representation for the model. We compare Hawkes process properties with their INAR($\infty$) counterparts. Given a Hawkes process $N$, in the main theorem of the paper we construct an INAR($\infty$)-based family of point processes and prove its convergence to $N$. This connection between INAR and Hawkes models will be relevant in applications.
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