Functional limit theorems for nonstationary marked Hawkes processes in the high intensity regime

Li, Bo; Pang, Guodong

doi:10.1016/j.spa.2021.10.008

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2022

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“…Hawkes processes have been widely used to model various applications, for example, in finance [2,11] and internet traffic and queueing [4,6,7,14]. There are extensive studies of Hawkes processes, including the exact distributional properties [12,16] for exponential-type kernel functions, and limit theorems in both conventional scaling [1,2,4,13] and large-intensity scaling [7,8,15].…”

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confidence: 99%

On the splitting and aggregating of Hawkes processes

Pang

2022

J. Appl. Probab.

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We consider the random splitting and aggregating of Hawkes processes. We present the random splitting schemes using the direct approach for counting processes, as well as the immigration–birth branching representations of Hawkes processes. From the second scheme, it is shown that random split Hawkes processes are again Hawkes. We discuss functional central limit theorems (FCLTs) for the scaled split processes from the different schemes. On the other hand, aggregating multivariate Hawkes processes may not necessarily be Hawkes. We identify a necessary and sufficient condition for the aggregated process to be Hawkes. We prove an FCLT for a multivariate Hawkes process under a random splitting and then aggregating scheme (under certain conditions, transforming into a Hawkes process of a different dimension).

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