On the splitting and aggregating of Hawkes processes

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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).

Original languageEnglish (US)
Pages (from-to)676-692
Number of pages17
JournalJournal of Applied Probability
Volume60
Issue number2
DOIs
StatePublished - 2023

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • General Mathematics
  • Statistics, Probability and Uncertainty

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