{"title":"Gaussian Approximation and Moderate Deviations of Poisson Shot Noises with Application to Compound Generalized Hawkes Processes","authors":"Mahmoud Khabou, Giovanni Luca Torrisi","doi":"arxiv-2409.06394","DOIUrl":null,"url":null,"abstract":"In this article, we give explicit bounds on the Wasserstein and the\nKolmogorov distances between random variables lying in the first chaos of the\nPoisson space and the standard Normal distribution, using the results proved by\nLast, Peccati and Schulte. Relying on the theory developed in the work of\nSaulis and Statulevicius and on a fine control of the cumulants of the first\nchaoses, we also derive moderate deviation principles, Bernstein-type\nconcentration inequalities and Normal approximation bounds with Cram\\'er\ncorrection terms for the same variables. The aforementioned results are then\napplied to Poisson shot-noise processes and, in particular, to the generalized\ncompound Hawkes point processes (a class of stochastic models, introduced in\nthis paper, which generalizes classical Hawkes processes). This extends the\nrecent results availale in the literature regarding the Normal approximation\nand moderate deviations.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"28 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.06394","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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 by
Last, Peccati and Schulte. Relying on the theory developed in the work of
Saulis and Statulevicius 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 availale in the literature regarding the Normal approximation
and moderate deviations.