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 satisfying $\P (f>t)t^{1/\alpha }\to c>0$ as $t\to \infty$, in exponentially mixing dynamical systems. We prove trimmed strong laws of large numbers when $\alpha \ge 1$, extending known results from the i.i.d.\ case. We further establish distributional limit theorems for both lightly and intermediately trimmed sums in the regime $\alpha >1/2$, showing convergence to an explicitly described non-standard law and to a normal distribution, respectively. The proofs rely on approximating trimmed sums by truncated ergodic sums and exploiting the system's exponential mixing properties.

arXiv comment59 pages, no figure, comments welcome