  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+\alpha) with 1/2 < \alpha < 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.
