We generalize Lindeberg’s proof of the central limit theorem to an invariance principle for arbitrary smooth functions of independent and weakly dependent random variables. The result is applied to get a similar theorem for smooth functions of exchangeable random variables. This theorem allows us to identify, for the first time, the limiting spectral distributions of Wigner matrices with exchangeable entries.
Sourav Chatterjee. "A generalization of the Lindeberg principle." Ann. Probab. 34 (6) 2061 - 2076, November 2006. https://doi.org/10.1214/009117906000000575