Limit theorems for a discrete-time marked Hawkes process

Wang, Haixu

doi:10.1016/j.spl.2022.109368

Cited by 5 publications

(2 citation statements)

References 19 publications

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“…The following are the proofs of the first main theorems. Note that Az = -(qp)z + pν, (31) where qp > 0. Using the definition of Z t process, the Foster-Lyapunov criterion (see [9] for details), and (i) of Assumption 1.1, we conclude that Z t is ergodic.…”

Section: Proof Of the Law Of Large Numbersmentioning

confidence: 99%

“…Seol [25] proposed a 0-1 discrete Hawkes process starting from empty history and proved some limit behaviors such as the law of large numbers (LLN), the invariance principles, and the central limit theorem (CLT). Recently, Wang [31,32] studied limit behaviors of a discrete-time Hawkes process with random marks and proved large and moderate deviations for a discrete-time Hawkes process with marks. Seol [27,28] studied the moderate deviation principle of marked Hawkes processes and also studied asymptotic behaviors for the compensator processes of Hawkes models.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic results for a class of Markovian self-exciting processes

Seol

2023

J Inequal Appl

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Hawkes process is a class of self-exciting point processes with clustering effect whose jump rate relies on their entire past history. This process is usually defined as a continuous-time setting and has been widely applied in several fields, including insurance, finance, queueing theory, and statistics. The Hawkes model is generally non-Markovian because the future development of a self-exciting point process is determined by the timing of past events. However, it can be Markovian in special cases such as when the exciting function is an exponential function or a sum of exponential functions. Difficulty arises when the exciting function is not an exponential function or a sum of exponentials, in which case the process can be non-Markovian. The inverse Markovian case for Hawkes processes was introduced by Seol (Stat. Probab. Lett. 155:108580, 2019) who studied some asymptotic behaviors. An extended version of the inverse Markovian Hawkes process was also studied by Seol (J. Korean Math. Soc. 58(4):819–833, 2021). In the current work, we propose a class of Markovian self-exciting processes that interpolates between the Hawkes process and the inverse Hawkes process. We derived limit theorems for the newly considered class of Markovian self-exciting processes. In particular, we established both the law of large numbers (LLN) and central limit theorems (CLT) with some key results.

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Section: Proof Of the Law Of Large Numbersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Asymptotic results for a class of Markovian self-exciting processes

Seol

2023

J Inequal Appl

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Study of discrete-time Hawkes process and its compensator

Sarma,

Selvamuthu

2024

Statistics & Probability Letters

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Non-Markovian Inverse Hawkes Processes

Seol

2022

Mathematics

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Hawkes processes are a class of self-exciting point processes with a clustering effect whose jump rate is determined by its past history. They are generally regarded as continuous-time processes and have been widely applied in a number of fields, such as insurance, finance, queueing, and statistics. The Hawkes model is generally non-Markovian because its future development depends on the timing of past events. However, it can be Markovian under certain circumstances. If the exciting function is an exponential function or a sum of exponential functions, the model can be Markovian with a generator of the model. In contrast to the general Hawkes processes, the inverse Hawkes process has some specific features and self-excitation indicates severity. Inverse Markovian Hawkes processes were introduced by Seol, who studied some asymptotic behaviors. An extended version of inverse Markovian Hawkes processes was also studied by Seol. With this paper, we propose a non-Markovian inverse Hawkes process, which is a more general inverse Hawkes process that features several existing models of self-exciting processes. In particular, we established both the law of large numbers (LLN) and Central limit theorems (CLT) for a newly considered non-Markovian inverse Hawkes process.

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Limit theorems for a discrete-time marked Hawkes process

Cited by 5 publications

References 19 publications

Asymptotic results for a class of Markovian self-exciting processes

Asymptotic results for a class of Markovian self-exciting processes

Study of discrete-time Hawkes process and its compensator

Non-Markovian Inverse Hawkes Processes

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