Trimmed strong laws and distributional limits for exponentially mixing systems

Source abstract

Abstract The Birkhoff ergodic theorem establishes pointwise convergence for integrable observables, but for f ∉ L 1 $f\notin L^1$ f not an element of upper L Superscript 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 / α → c > 0 $\mathbb {P}(f>t)\,t^{1/\alpha }\to c>0$ double struck upper P left parenthesis f greater than t right parenthesis t Superscript 1 divided by alpha Baseline right arrow c greater than 0 as t → ∞ $t\to \infty$ t right arrow infinity , in exponentially mixing dynamical systems. We prove trimmed strong laws of large numbers when α ≥ 1 $\alpha \geq 1$ alpha greater than or equals 1 , extending known results from the independent and identically distributed case. We further establish distributional limit theorems for both lightly and intermediately trimmed sums in the regime α > 1 / 2 $\alpha>1/2$ alpha greater than 1 divided by 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.