In dynamical systems a transformation $T$ is strongly mixing if $\lim_{n\rightarrow \infty} P(A \cap T^{-n} B) = P(A)P(B)$ (e.g., Patrick Billingsley's Ergodic Theory and Information)
For stochastic processes, mixing is usually defined differently through mixing coefficients, e.g. $\alpha$-mixing: https://en.wikipedia.org/wiki/Mixing_(mathematics)#Mixing_in_stochastic_processes
Since a stochastic process can also be modeled as a dynamical system (as opposed to a sequence of random variables), I'm wondering how the notion of mixing in dynamical systems is related to that in statistics. In particular, is $\alpha$-mixing coefficient simply the rate at which $|P(A \cap T^{-n} B)- P(A)P(B)|$ converges to zero?
The $\alpha$-mixing implies mixing in the statistical (ergodic-theoretical) sense, but not the other way round.
It is nicely explained in the Samorodnitsky's book "Stochastic processes and long range dependence".