dynamical systems: for piecewise monotonic maps by Hofbauer-Keller [@HK], for Anosov maps by Denker-Philipp [@DP] and for partially and non-uniformly hyperbolic systems by Dolgopyat [@Do] and Melbourne-Nicol [@MN]. We refer to the survey articles of Chernov [@C] and Denker [@De] for limit theorems and statistical properties concerning dynamical systems.\
The ASIP implies the CLT. Nevertheless, the latter can be directly proved via coding techniques and Ibragimov’s theorem [@I]. That method was employed by Sinai [@Sin] and Ratner [@R] for the geodesic flow in negative curvature, and by Bowen [@Bo] for Anosov maps. In the present article, Ibragimov’s condition is fulfilled by theorem B.\
The Gordin’s theorem provides another method for proving the CLT (see [@G], [@Li]). It relies on an approximation of $(\psi \circ
f^j)_{j \geq 0}$ by a sequence of reverse martingale differences. In our context, this can be done if $\sum_{n \geq 0} \norm{\Lambda^n
   \psi}_2$ (denoted $(\star)$) converges, where $\Lambda$ denotes the Ruelle-Perron-Frobenius operator (we have $\Lambda^n \psi (z) = {1 \over d_t^n}
 \sum_{y \in f^{-n}(z)} \psi(y)$ for every $z \in {\mathbb {P}}^k$). Let us note that the reverse martingale mentioned is defined with respect to the decreasing filtration $(f^{-n}\BB)_{n \geq 0}$, where $\BB$ is the Borel ${\sigma}$-algebra 