Editing Central limit theorem (section)

== The generalized central limit theorem ==

The generalized central limit theorem (GCLT) was an effort of multiple mathematicians ([[Sergei Natanovich Bernstein|Bernstein]], [[Jarl Waldemar Lindeberg|Lindeberg]], [[Paul Lévy (mathematician)|Lévy]], [[William Feller|Feller]], [[Andrey Kolmogorov|Kolmogorov]], and others) over the period from 1920 to 1937.<ref>{{cite journal |last1=Le Cam |first1=L. |title=The Central Limit Theorem around 1935 |journal=Statistical Science |date=February 1986 |volume=1 |issue=1 |pages=78–91 |jstor=2245503}}</ref> The first published complete proof of the GCLT was in 1937 by [[Paul Lévy (mathematician)|Paul Lévy]] in French.<ref>{{cite book |last1=Lévy |first1=Paul |title=Theorie de l'addition des variables aleatoires |lang=fr |trans-title=Combination theory of unpredictable variables |date=1937 |publisher=Gauthier-Villars |location=Paris}}</ref> An English language version of the complete proof of the GCLT is available in the translation of [[Boris Vladimirovich Gnedenko|Gnedenko]] and [[Andrey Kolmogorov|Kolmogorov]]'s 1954 book.<ref>{{cite book |last1=Gnedenko |first1=Boris Vladimirovich |last2=Kologorov |first2=Andreĭ Nikolaevich |last3=Doob |first3=Joseph L. |last4=Hsu |first4=Pao-Lu |title=Limit distributions for sums of independent random variables |date=1968 |publisher=Addison-wesley |location=Reading, MA}}</ref>

The statement of the GCLT is as follows:<ref>{{cite book |last1=Nolan |first1=John P. |title=Univariate stable distributions, Models for Heavy Tailed Data |series=Springer Series in Operations Research and Financial Engineering |date=2020 |publisher=Springer |location=Switzerland |doi=10.1007/978-3-030-52915-4 |isbn=978-3-030-52914-7 |s2cid=226648987 |url=https://doi.org/10.1007/978-3-030-52915-4}}</ref>

:''A non-degenerate random variable'' ''Z'' ''is [[stable distribution|''α''-stable]] for some'' 0 < ''α'' ≤ 2 ''if and only if there is an independent, identically distributed sequence of random variables'' ''X''<sub>1</sub>, ''X''<sub>2</sub>, ''X''<sub>3</sub>, ''... and constants'' ''a''<sub>''n''</sub> > 0, ''b''<sub>''n''</sub> ∈ ℝ ''with''

::''a''<sub>''n''</sub> (''X''<sub>1</sub> + ... + ''X''<sub>''n''</sub>) − ''b''<sub>''n''</sub> → ''Z''.

:''Here → means the sequence of random variable sums converges in distribution; i.e., the corresponding distributions satisfy'' ''F''<sub>''n''</sub>(''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]].