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I am looking for a proof based on characteristic functions for the generalized central limit theorem when the second moment does not exist, in which case one ends up with a power law rather than a Gaussian state.

This Theorem is described on the wikipedia page of the CLT click me but unfortunately there is no modern reference to this result given.

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Welcome to MO! This theorem is e.g. Theorem 14 on page 91 of Petrov's book, with further references to [24], [50].

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