A Central Limit Theorem for non-commutative random variables is proved using the Lindeberg method. The theorem is a generalization of the Central Limit Theorem for free random variables proved by Voiculescu. The Central Limit Theorem in this paper relies on an assumption which is weaker than freeness.
"A Proof of a Non-Commutative Central Limit Theorem by the Lindeberg Method." Electron. Commun. Probab. 12 36 - 50, 2007. https://doi.org/10.1214/ECP.v12-1250