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Trimmed strong laws and distributional limits for exponentially mixing systems

Max Auer, Sixu Liu

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 $¶(f>t)\asymp t^{-1/α}$ 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.

Trimmed strong laws and distributional limits for exponentially mixing systems

Abstract

The Birkhoff Ergodic Theorem establishes pointwise convergence for integrable observables, but for , no normalization yields almost sure convergence. This paper investigates trimmed ergodic sums, where the largest observations are removed, for observables with polynomial tails in exponentially mixing dynamical systems. We prove trimmed strong laws of large numbers when , 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 , 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.
Paper Structure (13 sections, 31 theorems, 326 equations)

This paper contains 13 sections, 31 theorems, 326 equations.

Table of Contents

  1. Results
  2. Introduction
  3. Polynomial Singularities
  4. Proofs
  5. Large observations and closest points
  6. Truncation
  7. Lightly trimmed SLLN
  8. Intermediately trimmed SLLN
  9. Returns to small balls
  10. Summation over Poisson point processes
  11. Lightly trimmed distributional limits
  12. Facts about distributional convergence
  13. Intermediately trimmed distributional limits

Key Result

Theorem 4

Assume that the system $(M, \mathcal{B}(M), \mu, T)$ is exponentially mixing. Let $f$ be a function with a power singularity of order $\operatorname{Ord}_{x^*}(f) = d$ at a slowly recurrent point $x^* \in M$, and fix $K \geq 1$. Then it holds that

Theorems & Definitions (67)

  • Definition 1
  • Remark 2
  • Definition 3: Slowly recurrent point
  • Theorem 4
  • Corollary 5: Convergence in probability
  • Theorem 6
  • Remark 7
  • Remark 8
  • Theorem 9
  • Remark 10
  • ...and 57 more