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](https://en.wikipedia.org/wiki/Central_limit_theorem#Generalized_theorem)
but unfortunately there is no modern reference to this result given.



Heavy tail central limit theorem
Welcome to MO! This theorem is e.g. [Theorem 14 on page 91 of Petrov's book](https://books.google.com/books/about/Sums_of_Independent_Random_Variables.html?id=zSDqCAAAQBAJ&printsec=frontcover&source=kp_read_button#v=onepage&q&f=false), with further references to [24], [50].


