If $X_{i}$ are a bunch of iid random variables with mean 0 and finite second moments, we know that $\sum_{i=1}^{n} \frac{X_{i}}{\sqrt{n}}$ converges in law to a Gaussian. Furthermore, by the Berry-Esseen theorem, we have some bounds on the rates of this convergence. Similar results hold, even if $X_{i}$ are only mixing in some sense, rather than actually iid.

If the $X_{i}$ are now instead regularly varying with exponent 1 (and satisfying some other conditions), and are 'very strongly mixing' but not iid, then I know that their suitably scaled partial sums converge in law to an appropriate Levy function with exponent 1.

My question is, what is a reasonable rate for this convergence? Of course, it will have something to do with the mixing rate, and with the constants associated with regular variation, and maybe other things. Are there any standard results in this direction?

I am not much of an expert, and have so far tried only some very standard tricks (e.g. concentration of regeneration times) which work in very specialized circumstances, and often not very well. Any pointers would be appreciated.

Edited: In response to Igor's very nice answer below, to state explicitly that I did not wish to assume the $X_{i}$ were iid.