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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.

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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.

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.

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