An analogue of Law of Iterated Logarithm for Heavy Tailed Random Variables
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866909761374519296 |
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| author | Dolgopyat, Dmitry Liu, Sixu |
| author_facet | Dolgopyat, Dmitry Liu, Sixu |
| contents | We establish functional limit theorems for ergodic sums of observables with power singularities for expanding circle maps. In the regime where the observables have infinite variance, we show that when rescaled by $N^{1/s}(\ln N)^α$, the partial sum process has limit points consisting precisely of increasing piecewise constant functions with finitely many jumps. Our approach combines trimming techniques with a multiple Borel-Cantelli argument. It provides a functional law of the iterated logarithm for heavy-tailed processes where classical almost sure invariance principles do not apply. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_15378 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An analogue of Law of Iterated Logarithm for Heavy Tailed Random Variables Dolgopyat, Dmitry Liu, Sixu Dynamical Systems Probability We establish functional limit theorems for ergodic sums of observables with power singularities for expanding circle maps. In the regime where the observables have infinite variance, we show that when rescaled by $N^{1/s}(\ln N)^α$, the partial sum process has limit points consisting precisely of increasing piecewise constant functions with finitely many jumps. Our approach combines trimming techniques with a multiple Borel-Cantelli argument. It provides a functional law of the iterated logarithm for heavy-tailed processes where classical almost sure invariance principles do not apply. |
| title | An analogue of Law of Iterated Logarithm for Heavy Tailed Random Variables |
| topic | Dynamical Systems Probability |
| url | https://arxiv.org/abs/2312.15378 |