In Bradley (1986, p. 170), it is stated that rho-mixing implies strong mixing. In White's book on asymptotic theory (p. 50, T3.49 and P3.50), it is stated that if $Z_t$, $X_t$ and $\epsilon_t$ are strong mixing, then measurable functions of these variables are strong mixing too. For lack of a better description, let us call this property transitivity.
My question is: does this mean that rho-mixing sequences are also transitive? My guess is that the answer is affirmative, but a conclusive answer would be useful.
Follow-up question: Peligrad (1986, p. 209-211) gives a CLT for rho-mixing second-order stationary random variables. Is it possible that this is the weakest CLT for covariance stationary processes out there?