Trimmed strong laws and distributional limits for exponentially mixing systems

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.