The CLT for rotated ergodic sums and related processes

Guy Cohen (1) , Jean-Pierre Conze (2)

1 Department of Electrical and Computer Engineering [Beer-Sheva]
2 IRMAR - Institut de Recherche Mathématique de Rennes

Let (Omega, A, P, tau) be an ergodic dynamical system. The rotated ergodic sums of a function f on Omega for theta is an element of R are S-n(theta) f := Sigma(n-1)(k=0) e(2 pi ik theta) for o tau(k) n >= 1. Using Carleson's theorem on Fourier series, Peligrad and Wu proved in [14] that (S(n)(theta)f)(n) (>=) (1) satisfies the CLT for a.e. theta when (f circle 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 Z(d)-dynamical systems.

Mots clés: Central Limit Theorem; rotated process; K-systems

Domaines: Probabilités [math.PR]

Guy Cohen, Jean-Pierre Conze. The CLT for rotated ergodic sums and related processes. Discrete and Continuous Dynamical Systems - Series A, 2013, 33 (9), pp.3981-4002. ⟨10.3934/dcds.2013.33.3981⟩. ⟨hal-00821919⟩