The CLT for rotated ergodic sums and related processes
Year: 2013
Type: article
Abstract: Let $( Ω , Α , \mathbb{P} , \tau )$ be an ergodic dynamical system.The rotated ergodic sums of a function $f$ on $\Omega$ for $\theta\in \mathbb{R}$ are $S_n^θ f : = \sum_{k=0}^{n-1} e^{2\pi i k \theta} f\circ \tau^k, n \geq 1$. Using Carleson's theorem on Fourierseries, Peligrad and Wu proved in [14] that $(S_n^\thetaf)_{n \geq 1}$ satisfies the CLT for a.e. $\theta$ when $(f\circ\tau^n)$ is a regular process. Our aim is to extend this result and give a simple proof based onthe Fejér-Lebesgue theorem. The results are expressed in theframework of processes generated by $K$-systems. We also considerthe invariance principle for modified rotated sums. In a lastsection, we extend the method to $\mathbb{Z}^d$-dynamical systems.
Authors: Guy Cohen, Jean-Pierre Conze
Institution: Institut de recherche mathématique de Rennes
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2321849697