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 $\{n:\mathrm{\mid }(T^{n}f,f)\mathrm{\mid }>\psi (n)\}$ is infinite. The mildly mixing automorphisms $T$ do not have convergences of non-zero averages $\frac{1}{k_n}{\sum }_{i=1}^{k_n}{T}^{i}f(x)$ with the rate of $o\left(\frac{1}{k_n}\right)$.