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 $X n$ , $n \in \mathbb{N}$ , be a sequence of independent, random variables. Suppose that each $X n$ has finite expected value $\mathbb{E} [X_{n}] = \mu_{n}$ and finite variance $\mathrm{Var} [X_{n}] = \sigma_{n}^{2}$ . Suppose also that the third central moments

$r_{n}^{3} := \mathbb{E} [ \left| X_{n} - \mu_{n} |^{3} \right]$

are finite and satisfy the Lyapunov condition

$\lim_{N \to \infty} \frac{\left( \sum_{n = 1}^{N} r_{n}^{3} \right)^{1/3}}{\left( \sum_{n = 1}^{N} \sigma_{n}^{2} \right)^{1/2}} = 0.$

Let the random variable $S_{N} := X_{1} + \dots + X_{N}$ denote the $N$ th partial sum of the random variables $X n$ . Then the normalised partial sum