Electronic Journal of Probability

Fluctuations for mean-field interacting age-dependent Hawkes processes

Julien Chevallier

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Abstract

The propagation of chaos and associated law of large numbers for mean-field interacting age-dependent Hawkes processes (when the number of processes $n$ goes to $+\infty $) being granted by the study performed in [9], the aim of the present paper is to prove the resulting functional central limit theorem. It involves the study of a measure-valued process describing the fluctuations (at scale $n^{-1/2}$) of the empirical measure of the ages around its limit value. This fluctuation process is proved to converge towards a limit process characterized by a limit system of stochastic differential equations driven by a Gaussian noise instead of Poisson (which occurs for the law of large numbers limit).

Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 42, 49 pp.

Dates
Received: 7 November 2016
Accepted: 27 April 2017
First available in Project Euclid: 3 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1493777018

Digital Object Identifier
doi:10.1214/17-EJP63

Mathematical Reviews number (MathSciNet)
MR3646068

Zentralblatt MATH identifier
1364.60062

Subjects
Primary: 60G55: Point processes 60F05: Central limit and other weak theorems 60G57: Random measures 60H15: Stochastic partial differential equations [See also 35R60] 92B20: Neural networks, artificial life and related topics [See also 68T05, 82C32, 94Cxx]

Keywords
Hawkes process central limit theorem interacting particle systems stochastic partial differential equation neural network

Rights
Creative Commons Attribution 4.0 International License.

Citation

Chevallier, Julien. Fluctuations for mean-field interacting age-dependent Hawkes processes. Electron. J. Probab. 22 (2017), paper no. 42, 49 pp. doi:10.1214/17-EJP63. https://projecteuclid.org/euclid.ejp/1493777018


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