Limiting Behavior of Maxima under Dependence
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917657301745664 |
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| author | Herrmann, Klaus Hofert, Marius Neslehova, Johanna G. |
| author_facet | Herrmann, Klaus Hofert, Marius Neslehova, Johanna G. |
| contents | Weak convergence of maxima of dependent sequences of identically distributed continuous random variables is studied under normalizing sequences arising as subsequences of the normalizing sequences from an associated iid sequence. This general framework allows one to derive several generalizations of the well-known Fisher-Tippett-Gnedenko theorem under conditions on the univariate marginal distribution and the dependence structure of the sequence. The limiting distributions are shown to be compositions of a generalized extreme value distribution and a distortion function which reflects the limiting behavior of the diagonal of the underlying copula. Uniform convergence rates for the weak convergence to the limiting distribution are also derived. Examples covering well-known dependence structures are provided. Several existing results, e.g. for exchangeable sequences or stationary time series, are embedded in the proposed framework. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02833 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Limiting Behavior of Maxima under Dependence Herrmann, Klaus Hofert, Marius Neslehova, Johanna G. Probability Statistics Theory 60G70, 62H05 Weak convergence of maxima of dependent sequences of identically distributed continuous random variables is studied under normalizing sequences arising as subsequences of the normalizing sequences from an associated iid sequence. This general framework allows one to derive several generalizations of the well-known Fisher-Tippett-Gnedenko theorem under conditions on the univariate marginal distribution and the dependence structure of the sequence. The limiting distributions are shown to be compositions of a generalized extreme value distribution and a distortion function which reflects the limiting behavior of the diagonal of the underlying copula. Uniform convergence rates for the weak convergence to the limiting distribution are also derived. Examples covering well-known dependence structures are provided. Several existing results, e.g. for exchangeable sequences or stationary time series, are embedded in the proposed framework. |
| title | Limiting Behavior of Maxima under Dependence |
| topic | Probability Statistics Theory 60G70, 62H05 |
| url | https://arxiv.org/abs/2405.02833 |