On the Glivenko-Cantelli theorem for real-valued empirical functions of stationary $α$-mixing and $β$-mixing sequences
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913813432893440 |
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| author | Coulibaly, Ousmane Sangaré, Harouna |
| author_facet | Coulibaly, Ousmane Sangaré, Harouna |
| contents | In this paper we extend the classical Glivenko-Cantelli theorem to real-valued empirical functions under dependence structures characterised by $α$-mixing and $β$-mixing conditions. We investigate sufficient conditions ensuring that families of real-valued functions exhibit the Glivenko-Cantelli (GC) property in these dependence settings. Our analysis focuses on function classes satisfying uniform entropy conditions and establishes deviation bounds under mixing coefficients that decay at appropriate rates. Our results refine the existing literature by relaxing the independence assumptions and highlighting the role of dependence in empirical process convergence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_20206 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Glivenko-Cantelli theorem for real-valued empirical functions of stationary $α$-mixing and $β$-mixing sequences Coulibaly, Ousmane Sangaré, Harouna Statistics Theory 62G20, 62G30, 60F99, 60G1 In this paper we extend the classical Glivenko-Cantelli theorem to real-valued empirical functions under dependence structures characterised by $α$-mixing and $β$-mixing conditions. We investigate sufficient conditions ensuring that families of real-valued functions exhibit the Glivenko-Cantelli (GC) property in these dependence settings. Our analysis focuses on function classes satisfying uniform entropy conditions and establishes deviation bounds under mixing coefficients that decay at appropriate rates. Our results refine the existing literature by relaxing the independence assumptions and highlighting the role of dependence in empirical process convergence. |
| title | On the Glivenko-Cantelli theorem for real-valued empirical functions of stationary $α$-mixing and $β$-mixing sequences |
| topic | Statistics Theory 62G20, 62G30, 60F99, 60G1 |
| url | https://arxiv.org/abs/2502.20206 |