Lyapunov's central limit theorem

Lyapunov's central limit theorem

In probability theory, Lyapunov's central limit theorem is one of the variants of the central limit theorem. Unlike the classical central limit theorem, which requires that the random variables in question be both independent and identically distributed, it only requires that they be independent. It is named after the Russian mathematician Aleksandr Lyapunov.

tatement 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} ight]

are finite and satisfy the Lyapunov condition

:lim_{N o infty} frac{left( sum_{n = 1}^{N} r_{n}^{3} ight)^{1/3{left( sum_{n = 1}^{N} sigma_{n}^{2} ight)^{1/2 = 0.

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

:Z_{N} := frac{S_{N} - sum_{n = 1}^{N} mu_{n{left( sum_{n = 1}^{N} sigma_{n}^{2} ight)^{1/2

converges in distribution to a standard normal random variable as N o infty.

Less formally, for "large" N, S_{N} is approximately normally distributed with expected value

:mathbb{E} [S_{N}] approx sum_{n = 1}^{N} mathbb{E} [X_{n}]

and variance

:mathrm{Var} [S_{N}] approx sum_{n = 1}^{N} mathrm{Var} [X_{n}] .

External links

* [http://www.stat.sfu.ca/~lockhart/richard/801/00_1/lectures/09/web.html Central limit theorems]


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