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.



Generalized central limit theorem
One of the most recent results in this area is 


Katarzyna Bartkiewicz, Adam Jakubowski, Thomas Mikosch, Olivier Wintenberger, *Stable limits for sums of dependent infinite variance random variables*, Probability Theory and Related Fields 150, 3 (2011) 337-372


which is available [here](http://arxiv.org/abs/0906.2717).


It gives sufficient conditions on $(X\_j)$ in order to ensure the convergence of partial sums divided by $a\_n$, where the conditions involve $a\_n$. It seems these ones involve $\alpha$-mixing coefficients. Work on $\phi$-mixing coefficient has been performed in 


Thomas Mikosch, Daniel Straumann, *Stable limits of martingale transforms with application to the estimation of GARCH parameters*, Annals of Statistics 2006, Vol. 34, No. 1, 493-522,


which can be downloaded [here](http://arxiv.org/abs/math/0605613).


