Limit of sums of iid random variables which are not square-integrable

The generalized central limit theorem states (see this for a summary), that if $X_i$ are i.i.d. such that its density function has left tail power-law asymptotic $\mathbb{P}(X < -x) \sim d x^{-\mu}$ and right tail asymptotic $\mathbb{P}(X > x) \sim 1- c x^{-\mu}$ as $x \to +\infty$, then there exist sequences $a_n$ and $b_n$ such that the random variate $Z_n = ((\sum_{i=1}^n X_i) - a_n )/b_n$ converges in probability to a stable distribution with stability index $\alpha = \min(\mu, 2)$ and asymmetry parameter $\beta = \frac{c-d}{c+d}$.

Details on the constructive choice of sequences $a_n$ and $b_n$ are given in the table found at the link above. Also see page 62 of Zolotarev and Uchaikin on Google books.