We generalize a method developed by Sarig to obtain polynomial lower bounds for correlation functions for maps with a countable Markov partition. A consequence is that LS Young's estimates on towers are always optimal. Moreover, we show that, for functions with zero average, the decay rate is better, gaining a factor 1/n. This implies a Central Limit Theorem in contexts where it was not expected, e.g. x+Cx^(1+α) with 1/2 < α< 1. The method is based on a general result on renewal sequences of operator, and gives an asymptotic estimate up to any precision of such operators.

33 pages