• Journal of the Korean Mathematical Society
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• v.56 no.4
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• pp.935-946
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• 2019

Hawkes process is a self-exciting simple point process with clustering effect whose jump rate depends on its entire past history. We consider Hawkes processes with uniform immigrants which is a special case of the Hawkes processes with renewal immigrants. We study the limit theorems for Hawkes processes with uniform immigrants. In particular, we obtain a law of large number, a central limit theorem, and a large deviation principle.

• Journal of the Korean Mathematical Society
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• v.58 no.4
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• pp.819-833
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• 2021

Hawkes process is a self-exciting simple point process with clustering effect whose jump rate depends on its entire past history and has been widely applied in insurance, finance, queueing theory, statistics, and many other fields. Seol [27] proposed the inverse Markovian Hawkes processes and studied some asymptotic behaviors. In this paper, we consider an extended inverse Markovian Hawkes process which combines a Markovian Hawkes process and inverse Markovian Hawkes process with features of several existing models of self-exciting processes. We study the limit theorems for an extended inverse Markovian Hawkes process. In particular, we obtain a law of large number and central limit theorems.

• The Korean Journal of Applied Statistics
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• v.36 no.4
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• pp.309-322
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• 2023

The Hawkes process is a point process with self-exciting characteristics. It has been mainly used to describe seismic phenomena in which aftershocks occur due to the main earthquake. Recently, it has been used to explain various phenomena with self-exciting properties, such as the spread of infectious diseases and the spread of news on SNS. The Hawkes process can be flexibly modified according to the characteristics of events by using various types of excitation functions. Since it is difficult to implement a maximum likelihood estimator numerically, estimation methods have been improved until recently. In this paper, the conditional intensity function and excitation function are explained to describe the Hawkes process. Then, existing examples of Hawkes processes used in seismic, epidemiological, criminal, and financial fields are described and estimation methods are introduced. I analyze earthquakes that occurred in gyeongsang-do, Korea from November 2017 to December 2022, using R package ETAS.