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**Asymptotics for Hawkes processes with large and small baseline intensities.**
*(English)*
Zbl 1475.60093

Summary: This paper focuses on asymptotic results for linear Hawkes processes with large and small baseline intensities. The intensity process is one of the main tools used to work with the dynamical properties of a general point process. It is of essential interest in credit risk study, in particular. First, we establish a large deviation principle and a moderate deviation principle for the Hawkes process with large baseline intensity. In addition, a law of large numbers and a central limit theorem are also obtained. Second, we observe asymptotic behaviors for the Hawkes process with small baseline intensity. The main idea of the proof relies on the immigration-birth representation and the observations of the moment generating function for the linear Hawkes process.

### MSC:

60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |

60F05 | Central limit and other weak theorems |

60F10 | Large deviations |

### Keywords:

Hawkes processes; intensity process; central limit theorems; law of large numbers; large deviations; moderate deviations### References:

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