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 Fourier series, Peligrad and Wu proved in [14] that $(S_n^\theta f)_{n \geq 1}$ 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 on the Fejer-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 $\mathbb{Z}^d$-dynamical systems.
Highlights
To cite this version: Guy Cohen, Jean-Pierre Conze
Using Carleson’s theorem on Fourier series, Peligrad and Wu proved in [14] that (Snθ f )n≥1 satisfies the Central Limit Theorem (CLT) for a.e. θ when (f ◦ τ n) is a regular process
We extend the method to Zd-dynamical systems
Summary
To cite this version: Guy Cohen, Jean-Pierre Conze. The CLT for rotated ergodic sums and related processes. Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2013, 33 (9), pp.3981-4002. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés
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