The CLT for rotated ergodic sums and related processes

Let (Ω,A, P,τ) be an ergodic dynamical system. The rotated ergodic sums of a function f on Ω for θ ∈ ℝ are S _n^θf:= Σ_k=0^n-1 e ^2πikθ foτ^k, n ≥ 1. Using Carleson's theorem on Fourier series, Peligrad and Wu proved in [14] that (S _n^θf)_n≥1 satisfies the CLT for a.e. θ when (f oτⁿ) is a regular process. Our aim is to extend this result and give a simple proof based on the Fejér-Lebesgue theorem. The results are expressed in the framework of processes generated by K-systems. We also consider the invariance principle for modified rotated sums. In a last section, we extend the method to ℤ^d-dynamical systems.