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
with
![\operatorname{E}[X_{ni}] = 0, \quad \forall i,](a/../upload/math/4d97722cbe9e4ad85787865ab7816065.png)
and
![\operatorname{E}\left[\left|{X_{ni}}^3\right|\right] \rightarrow 0,\,](a/../upload/math/50d58b5c2ca5a5ce1e25c3b97b81e8c3.png)
then
converges in distribution, as n → ∞, to the standard normal distribution.
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