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 $o(1/{k_n})$.