Trimmed strong laws and distributional limits for exponentially mixing systems
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917198759460864 |
|---|---|
| author | Auer, Max Liu, Sixu |
| author_facet | Auer, Max Liu, Sixu |
| contents | The Birkhoff Ergodic Theorem establishes pointwise convergence for integrable observables, but for $f\notin L^1$, no normalization yields almost sure convergence. This paper investigates trimmed ergodic sums, where the largest observations are removed, for observables with polynomial tails $¶(f>t)\asymp t^{-1/α}$ in exponentially mixing dynamical systems. We prove trimmed strong laws of large numbers when $α\geq 1$, extending known results from the i.i.d.\ case. Moreover, we establish distributional limit theorems for both lightly and intermediately trimmed sums in the regime $α>1/2$, showing convergence to a non-standard law, which we describe explicitly, and a normal distribution, respectively. The proofs rely on approximating the trimmed sums by truncated ergodic sums and exploiting the system's exponential mixing properties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_08126 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Trimmed strong laws and distributional limits for exponentially mixing systems Auer, Max Liu, Sixu Dynamical Systems Primary: 37A50, 60F15, 60G70, Secondary: 37C40 The Birkhoff Ergodic Theorem establishes pointwise convergence for integrable observables, but for $f\notin L^1$, no normalization yields almost sure convergence. This paper investigates trimmed ergodic sums, where the largest observations are removed, for observables with polynomial tails $¶(f>t)\asymp t^{-1/α}$ in exponentially mixing dynamical systems. We prove trimmed strong laws of large numbers when $α\geq 1$, extending known results from the i.i.d.\ case. Moreover, we establish distributional limit theorems for both lightly and intermediately trimmed sums in the regime $α>1/2$, showing convergence to a non-standard law, which we describe explicitly, and a normal distribution, respectively. The proofs rely on approximating the trimmed sums by truncated ergodic sums and exploiting the system's exponential mixing properties. |
| title | Trimmed strong laws and distributional limits for exponentially mixing systems |
| topic | Dynamical Systems Primary: 37A50, 60F15, 60G70, Secondary: 37C40 |
| url | https://arxiv.org/abs/2601.08126 |