Random walks generated by area preserving maps with zero Lyapounov exponents

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, , 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.