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Trimmed ergodic sums for non-integrable functions with power singularities over irrational rotations

Max Auer, Tanja I. Schindler

TL;DR

The paper addresses the problem of Birkhoff sums for non-integrable observables under irrational rotations and the failure of pointwise limit theorems due to large observations. It develops trimming as a robust remedy and proves trimmed weak and strong laws for the non-integrable functions $f(x)=\frac{1}{x}$ and $f(x)=x^{-\beta}$ with $\beta>1$, under Diophantine conditions on the rotation number $\alpha$ (notably Roth-type), while providing explicit normalizers $d_N$ and precise trimming sequences $k(N)=o(N)$. A general existence result shows trimming sequences exist for any ergodic system, with uniform convergence under unique ergodicity for continuous observables, and rotation-specific results sharpen the rate and necessity of trimming. The findings extend trimmed-sums theory beyond mixing settings, connect to prior work on Gauss and doubling maps, and furnish sharp criteria for when trimming is essential versus when untrimmed laws suffice, highlighting the role of Diophantine properties of $\alpha$ in determining the limit behavior. The work contributes rigorous, quantitative trimming thresholds and normalizers in a deterministic dynamical systems setting, with implications for understanding heavy-tailed observations in near-integrable systems.

Abstract

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.

Trimmed ergodic sums for non-integrable functions with power singularities over irrational rotations

TL;DR

The paper addresses the problem of Birkhoff sums for non-integrable observables under irrational rotations and the failure of pointwise limit theorems due to large observations. It develops trimming as a robust remedy and proves trimmed weak and strong laws for the non-integrable functions and with , under Diophantine conditions on the rotation number (notably Roth-type), while providing explicit normalizers and precise trimming sequences . A general existence result shows trimming sequences exist for any ergodic system, with uniform convergence under unique ergodicity for continuous observables, and rotation-specific results sharpen the rate and necessity of trimming. The findings extend trimmed-sums theory beyond mixing settings, connect to prior work on Gauss and doubling maps, and furnish sharp criteria for when trimming is essential versus when untrimmed laws suffice, highlighting the role of Diophantine properties of in determining the limit behavior. The work contributes rigorous, quantitative trimming thresholds and normalizers in a deterministic dynamical systems setting, with implications for understanding heavy-tailed observations in near-integrable systems.

Abstract

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 . In this setting we establish trimmed weak and strong laws for the functions and with , providing explicit conditions on the rotation angle.

Paper Structure

This paper contains 16 sections, 35 theorems, 246 equations, 1 table.

Table of Contents

  1. Results
  2. Introduction
  3. Motivation
  4. Background
  5. General results
  6. Trimming for rotations
  7. Background
  8. The function $\frac{1}{x}$
  9. The function $x^{-\beta}$ for $\beta>1$
  10. Comparison of results to literature
  11. Proofs
  12. General trimmed laws
  13. Irrational rotations on $\mathbb{T}$
  14. Background
  15. The function $\frac{1}{x}$
  16. ...and 1 more sections

Key Result

Theorem 2

Let $(X, T, \mu)$ be a probability preserving ergodic system. Then, for $f\in \mathcal{F}$ there exist $k(N)\in \mathbb{N}$ with $k(N)=o(N)$ and $d_N>0$ such that Furthermore, if $(X, T, \mu)$ is uniquely ergodic (and $\mu$ is regular) and $f$ is continuous, then $k(N)$ and $d_N$ can be chosen such that convergence in genthmclaim holds uniformlyFor definiteness, whenever we talk about uniform con

Theorems & Definitions (77)

  • Remark 1
  • Theorem 2
  • Theorem 3
  • Definition 4
  • Theorem 5
  • Remark 6
  • Remark 7
  • Theorem 8
  • Theorem 9
  • Remark 10
  • ...and 67 more