Lyapunov's central limit theorem
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In probability theory, Lyapunov's central limit theorem is one of the variants of the central limit theorem. Unlike the classical central limit theorem, it does not require that the random variables in question be independent and identically distributed. It is named for the Russian mathematician Aleksandr Lyapunov.
[edit] Statement of the theorem
Let Xn, 
, be a sequence of independent, random variables. Suppose that each Xn has finite expected value ![\mathbb{E} [X_{n}] = \mu_{n}](../../../math/6/6/9/6691a9c05c3d336bfd89b4afa5c28035.png)
and finite variance ![\mathrm{Var} [X_{n}] = \sigma_{n}^{2}](../../../math/6/1/0/6107ffc942ad2db02ab3442915ec2f30.png)
. Suppose also that the third central moments
![r_{n}^{3} := \mathbb{E} [ \left| X_{n} - \mu_{n} |^{3} \right]](../../../math/8/9/2/892a7ab519c122ff31994d5af3bed9d4.png)
are finite and satisfy the Lyapunov condition
Let the random variable 
denote the Nth partial sum of the random variables Xn. Then the normalised partial sum
converges in distribution to a standard normal random variable as 
.
Less formally, for "large" N, SN is approximately normally distributed with expected value
![\mathbb{E} [S_{N}] \approx \sum_{n = 1}^{N} \mathbb{E} [X_{n}]](../../../math/1/d/b/1db8e234b8464c70f14b5e6d5a5734f8.png)
and variance
![\mathrm{Var} [S_{N}] \approx \sum_{n = 1}^{N} \mathrm{Var} [X_{n}].](../../../math/c/3/d/c3dae86046bbd235a073c04826f49dba.png)
