Conditions are given which imply that the partial sums of a sequence of independent integer-valued random variables, suitably normalized, converge in distribution to a stable law of exponent $\alpha, 0 < \alpha < 2$, and imply as well that a strong version of the corresponding local limit theorem holds.
"A Stable Local Limit Theorem." Ann. Probab. 2 (1) 167 - 172, February, 1974. https://doi.org/10.1214/aop/1176996764