Trimmed ergodic sums for non-integrable functions with power singularities over irrational rotations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908288397869056 |
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| author | Auer, Max Schindler, Tanja I. |
| author_facet | Auer, Max Schindler, Tanja I. |
| contents | Studying Birkhoff sums of non-integrable functions involves the challenge of large observations depending on the sampled orbit, which prevents pointwise limit theorems. To address this issue, the largest observations are removed, this process is commonly known as trimming. While this method is well studied for independent identically distributed sequences and systems with strong mixing behaviour, this paper focuses on irrational rotations of $\mathbb{T}$. In this setting we establish trimmed weak and strong laws for the functions $\frac{1}{x}$ and $\frac{1}{x^β}$ with $β>1$, providing explicit conditions on the rotation angle. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_22242 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Trimmed ergodic sums for non-integrable functions with power singularities over irrational rotations Auer, Max Schindler, Tanja I. Dynamical Systems Number Theory 37A50, 60F15, 37A44 Studying Birkhoff sums of non-integrable functions involves the challenge of large observations depending on the sampled orbit, which prevents pointwise limit theorems. To address this issue, the largest observations are removed, this process is commonly known as trimming. While this method is well studied for independent identically distributed sequences and systems with strong mixing behaviour, this paper focuses on irrational rotations of $\mathbb{T}$. In this setting we establish trimmed weak and strong laws for the functions $\frac{1}{x}$ and $\frac{1}{x^β}$ with $β>1$, providing explicit conditions on the rotation angle. |
| title | Trimmed ergodic sums for non-integrable functions with power singularities over irrational rotations |
| topic | Dynamical Systems Number Theory 37A50, 60F15, 37A44 |
| url | https://arxiv.org/abs/2503.22242 |