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The Generalized Central Limit Theorem[edit]
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. [1] The first published complete proof of the GCLT was in 1937 by Paul Lévy in French.[2] An English language version of the complete proof of the GCLT is available in the translation of Gnedenko and Kolmogorov's 1954 book.[3]
The statement of the GLCT is as follows:[4]
- 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.
- ^ Le Cam, L. (February 1986). "The Central Limit Theorem around 1935". Statistical Science. 1 (1): 78-91.
- ^ Lévy, Paul (1937). Theorie de l’addition des variables aleatoires [Combination theory of unpredictable variables]. Paris: Gauthier-Villars.
- ^ Gnedenko, Boris Vladimirovich; Kologorov, Andreĭ Nikolaevich; Doob, Joseph L.; Hsu, Pao-Lu (1968). Limit distributions for sums of independent random variables. Reading, MA: Addison-wesley.
- ^ Nolan, John P. (2020). Univariate stable distributions, Models for Heavy Tailed Data. Switzerland: Springer. ISBN 978-3-030-52914-7.