I am looking for a generalized central limit theorem for non-square integrable stationary sequences. More precisely I suspect that when $(X_j)_{j\geqslant 1}$ is a stationary sequence such that $X_i$ belongs to the domain of attraction of the $\alpha$-stable law then

$$n^{1/\alpha}\sum_{i=1}^n X_i$$

converges as soon as the dependence of $X_i$ is not too strong (maybe some mixing condition).

Are there results of this sort known? Or maybe I am wrong and my conjecture is false.