Slow decay of correlations for generic mixing automorphisms

Let $\psi(n)\to +0$ and a square-integrable function $f$ be non-zero, then for the typical mixing automorphism $T$ the set $\{n:\, |(T^nf,f) |>\psi( n)\}$ is infinite. The mildly mixing automorphisms $T$ do not have convergences of non-zero averages $\frac 1 {k_n} \sum_{i=1}^{k_n}T^if (x)$ with the rate of $o\left(\frac 1 {k_n}\right)$.