Abstract
We study the asymptotic limit distributions of Birkhoff sums S n of a sequence of random variables of dynamical systems with zero entropy and Lebesgue spectrum type. A dynamical system of this family is a skew product over a translation by an angle α. The sequence has long memory effects. It comes that when α/ π is irrational the asymptotic behavior of the moments of the normalized sums S n / f n depends on the properties of the continuous fraction expansion of α. In particular, the moments of order k, E((S n/ n ) k) , are finite and bounded with respect to n when α/ π has bounded continuous fraction expansion. The consequences of these properties on the validity or not of the central limit theorem are discussed.
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