We give a general method of deriving statistical limit theorems, such as central limit theorem and its functional version, in situations where the iterates of discrete time maps display a polynomial decay of correlations. Let Y be a measurable space, T: Y→Y be a (non-invertible) measurable transformation, and m be an invariant and ergodic (normalized) measure for T. One of our main results is the following:

If h is a square integrable function on Y, with zero integral, and the L² norm of the n-th iterate of the Frobenius-Perron operator for T decay polynomially as n^c with c<-1/2 then the Central Limit Theorem and the Weak Invariance Principle hold for the sequence h(Tⁿ).

This theory gives a unified approach when studying statistical properties of transformations either on intervals, manifolds, or metric spaces. It covers most known examples. In the specific case of maps with an indifferent fixed point such as the Manneville-Pomeau map the result gives the best possible answer (for functions for which the decay is slower the central limit theorem might not hold).