I would like to know if there is a notion or an example related to the following situation: a transform $T$ on a space $E$, which is equipped with a measure $\mu$, satisfies $\mu(A\cap T^{-n}B)\sim \mu(A)\mu(B)c_n$ as $n$ tends to infinity, for any measurable subets $A$ and $B$ in some finite measure subset of $E$, where $c_n$ is regularly varying with an index between -1 and 0.