Trimmed strong laws and distributional limits for exponentially mixing systems

TL;DR

This paper extends strong laws and distributional limit theorems for trimmed sums in exponentially mixing systems with heavy-tailed observables, revealing new asymptotic behaviors and explicit limit laws.

Contribution

It introduces novel results on trimmed ergodic sums for polynomial tail observables in exponentially mixing systems, including strong laws and explicit distributional limits.

Findings

Proves trimmed strong laws for $ ext{α} ext{≥} 1$

Establishes distributional limits for trimmed sums when $ ext{α} > 1/2$

Shows convergence to non-standard laws and normal distributions

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.