多尺度高强度机制下非平稳标记霍克斯过程的函数极限行为
Functional limiting behaviour of non-stationary marked Hawkes processes under multi-scaling high-intensity regime
- Technion–Israel Institute of Technology（以色列理工学院）
- Indian Institute of Technology Delhi（德里印度理工学院）
- Indian Institute of Technology Guwahati（古瓦哈提印度理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究非平稳标记霍克斯过程在高强度机制下的函数极限行为，建立了函数大数定律和中心极限定理，证明重标度过程收敛于高斯过程，并研究了散粒噪声过程。
AI中文摘要：
本文研究了非平稳标记霍克斯过程，其中基础强度函数是时变的，核函数由称为标记的外部随机因素控制。假设标记由随时间发生的随机事件决定，从而导致非同分布。这些事实带来了两类困难：一类源于缺乏独立同分布（i.i.d.），另一类源于非平稳性。在此框架下，考虑了一种通常称为高强度机制的渐近机制，在该机制下强度随时间增加。这可以通过将时间参数乘以n的alpha次方（对于某个alpha>0）来实现。在高强度机制下，建立了函数大数定律和函数中心极限定理。证明了重标度（中心化和归一化）的标记霍克斯过程在分布上收敛于高斯过程，该高斯过程是高斯噪声和扩散过程的累积。此外，还研究了散粒噪声过程，并在适当假设下建立了函数极限定理。
英文摘要：
This paper studies the non-stationary marked Hawkes processes in which the base intensity function is time-dependent and the kernel function is governed by an external random factor called mark. The marks are assumed to be determined by random events occurrence over time resulting in a non-identical distribution. These facts introduce two kinds of difficulties: one from the absence of the i.i.d. (independent and identical distributed) and the other is from the non-stationarity. In this framework, an asymptotic regime, often referred to as the high intensity regime, is considered, under which the intensity increases with time. This can be obtained by multiplying the time parameter by n alpha, for some alpha > 0. Under the high intensity regime, the functional law of large numbers and the functional central limit theorem are established. The rescaled (centered and normalized) marked Hawkes process is proved to converge in distribution to a Gaussian process, which is the accumulation of Gaussian noise and a diffusion process. In addition, a shot-noise process is studied, and the functional Limit Theorems under appropriate hypotheses are established.
