Generic Correlations and Ergodic Averages for Strongly and Mildly Mixing Automorphisms

Given a sequence $$\psi(n)\to +0$$ and a square integrable nonzero function $$f$$ , the set $$\{n:|(T^nf,f)|>\psi(n)\}$$ is infinite for any generic mixing automorphism $$T$$ . For mildly mixing automorphisms $$T$$ , the nonzero averages $$1/{k_n}\sum_{i=1}^{k_n}T^if (x)$$ do not converge at a rate of $$o(1/{k_n})$$ .