A measure preserving transformation is called strongly mixing if the correlation between a measurement done at time zero and a measurement done at time n goes to zero as n tends to infinity. Non-uniform hyperbolicity is expected to slow down this process for `smooth' measurements, but to prove that this actually happens one needs to estimate the correlations from below. I will describe (the first) general method to do this. The key idea is an operator-theoretic version of the renewal theorem.