dorsal/arxiv
View SchemaTrimmed strong laws and distributional limits for exponentially mixing systems
| Authors | Max Auer, Sixu Liu |
|---|---|
| Categories | |
| ArXiv ID | 2601.08126vv1 |
| URL | https://arxiv.org/abs/2601.08126 |
| License | http://creativecommons.org/licenses/by/4.0/ |
Abstract
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 $\P(f>t)\asymp t^{-1/\alpha}$ in exponentially mixing dynamical systems. We prove trimmed strong laws of large numbers when $\alpha\geq 1$, 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 $\alpha>1/2$, 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.
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"abstract": "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 $\\P(f\u003et)\\asymp t^{-1/\\alpha}$ in exponentially mixing dynamical systems. We prove trimmed strong laws of large numbers when $\\alpha\\geq 1$, 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 $\\alpha\u003e1/2$, 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\u0027s exponential mixing properties.",
"arxiv_id": "2601.08126",
"authors": [
"Max Auer",
"Sixu Liu"
],
"categories": [
"math.DS"
],
"license": "http://creativecommons.org/licenses/by/4.0/",
"title": "Trimmed strong laws and distributional limits for exponentially mixing systems",
"url": "https://arxiv.org/abs/2601.08126",
"version": "v1"
},
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