Terence Tao has mentioned the importance of the Lindeberg exchange strategy, citing as an application how it was used in the proofs of some recent results relating to universality laws for random matrices. In a presentation from 2009, he explains the strategy as follows:

- Prove the result holds for $F(X_1,\dots,X_n)$. Here $F$ is a "nice" function, and the random variables $X_i$ are iid Gaussian.
- Prove that the distribution of $F$ is invariant under replacement of the distribution of the random variables, or more precisely, that when $n$ grows, $F(X_1,\dots,X_n)$ has the same distribution asymptotically as $F(Y_1,\dots,Y_n)$, where the $Y_i$ are iid from some distribution $D$.

The original application of the exchange strategy seems to have been Lindeberg's proof of the Central Limit Theorem, where $F(X_1,\dots,X_n) = \sum_{i=1}^n X_i/\sqrt{n}$. (Tao discusses this example in slides 32-34.)

My question is:

Is it feasible to use the Lindeberg exchange strategy with a stable distribution that does not have finite variance?

It is possible that the Lindeberg exchange strategy essentially requires the finite variance assumption. Removing it certainly breaks the proofs I've looked at (although I am no expert and have not done an exhaustive survey). However, the overall strategy doesn't obviously require finite variance. Is there something fundamental about the exchange strategy that relies on Gaussians?