The Ultimate Lyapunov's central limit theorem - American History Information Guide and Reference

In probability theory, Lyapunov's central limit theorem is one of the variants of the central limit theorems. It states that if for each positive integer N we have a sequence of independent, mean 0 random variables

$X_{n1}, X_{n2}, \ldots, X_{nn}$

with

$\operatorname{E}[X_{ni}] = 0, \quad \forall i,$

$\operatorname{Var}\left(\sum_{i}X_{ni}\right) = 1,\,\!$

and

$\operatorname{E}\left[\left|{X_{ni}}^3\right|\right] \rightarrow 0,\,$

then

$\sum_{i}X_{ni}\,\!$

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Last updated: 08-01-2005 23:55:55