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On the splitting and aggregating of Hawkes processes

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  09 December 2022

Bo Li  and
Guodong Pang
Bo Li*
Affiliation:
Nankai University
Guodong Pang*
Affiliation:
Rice University
*
*Postal address: School of Mathematics and LPMC, Nankai University. Email address: libo@nankai.edu.cn
**Postal address: Department of Computational Applied Mathematics and Operations Research, George R. Brown College of Engineering, Rice University. Email address: gdpang@rice.edu
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Abstract

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).

Keywords

Hawes processrandom splitting/samplingaggregating/superpositionfunctional central limit theoremBrownian motion

MSC classification

Primary: 60G55: Point processes
Secondary: 60F17: Functional limit theorems; invariance principles
Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 2 , June 2023 , pp. 676 - 692
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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