Gaussian approximation and moderate deviations of poisson shot noises with application to compound generalized Hawkes processes
Author(s)
Khabou, Mahmoud
Torrisi, Giovanni Luca
Type
Journal Article
Abstract
In this article, we give explicit bounds on the Wasserstein and the Kolmogorov distances between random variables lying in the first chaos of the Poisson space and the standard Normal distribution, using the results proved in [30]. Relying on the theory developed in [40] and on a
fine control of the cumulants of the first chaoses, we also derive moderate deviation principles, Bernstein-type concentration inequalities and Normal approximation bounds with Cram´er correction terms for the same variables. The aforementioned results are then applied to Poisson shot-noise processes and, in particular, to the generalized compound Hawkes point processes
(a class of stochastic models, introduced in this paper, which generalizes classical Hawkes processes). This extends the recent results in [23, 26] regarding the Normal approximation and in [47] for moderate deviations.
fine control of the cumulants of the first chaoses, we also derive moderate deviation principles, Bernstein-type concentration inequalities and Normal approximation bounds with Cram´er correction terms for the same variables. The aforementioned results are then applied to Poisson shot-noise processes and, in particular, to the generalized compound Hawkes point processes
(a class of stochastic models, introduced in this paper, which generalizes classical Hawkes processes). This extends the recent results in [23, 26] regarding the Normal approximation and in [47] for moderate deviations.
Date Issued
2025-03-01
Date Acceptance
2024-06-04
Citation
Advances in Applied Probability, 2025, 57 (1), pp.305-345
ISSN
0001-8678
Publisher
Applied Probability Trust
Start Page
305
End Page
345
Journal / Book Title
Advances in Applied Probability
Volume
57
Issue
1
Copyright Statement
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
License URL
Identifier
10.1017/apr.2024.51
Subjects
Generalized Hawkes processes
moderate deviations
Wasserstein distance
Kolmogorov distance
concentration inequalities 2010 Mathematics Subject Classification: Primary 60G55
60F05
60F10 Secondary 60G57
60H05
Publication Status
Published
Date Publish Online
2024-12-02