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The Generalized Central Limit Theorem
The Generalized Central Limit Theorem (GCLT) was a an effort of multiple mathematicians (Berstein, Lindeberg, Lévy, Feller, Kolmogorov, and others) over the period from 1920 to 1937. The first published complete proof of the GCLT was in 1937 by Paul Lévy in French. An English language version of the complete proof of the GCLT is available in the translation of Gnedenko and Kolmogorov's 1954 book.

The statement of the GLCT is as follows:


 * ''A non-degenerate random variable Z is α-stable for some 0 < α ≤ 2 if and only if there is an independent, identically distributed sequence of random variables X1, X2, X3, ... and constants an > 0, bn ∈ ℝ with


 * ''an (X1 + ... + Xn) - bn → Z.


 * ''Here → means the sequence of random variable sums converges in distribution; i.e., the corresponding distributions satisfy Fn(y) → F(y) at all continuity points of F.

In other words, if sums of independent, identically distributed random variables converge in distribution to some Z, then Z must be a stable distribution.