Lindeberg’s central limit theorem

Theorem (Lindeberg’s central limit theorem)

Let $X_1,X_2,\mathrm{\dots }$ be independent random variables with distribution functions $F_1,F_2,\mathrm{\dots }$, respectively, such that $E{X}_n={\mu }_n$ and , with at least one ${\sigma }_n>0$. Let

$$S_n=X_1+\mathrm{\cdots }+X_n\text{and}{s}_n=\sqrt{\mathrm{Var}(S_n)}=\sqrt{{\sigma }_1^2+\mathrm{\cdots }+{\sigma }_n^2}.$$

Then the normalized partial sums $\frac{S_n-E{S}_n}{s_n}$ converge in distribution (http://planetmath.org/ConvergenceInDistribution) to a random variable with normal distribution $N(0,1)$ (i.e. the normal convergence holds,) if the following Lindeberg condition is satisfied:

$$\forall \epsilon >0,\underset{n\to \mathrm{\infty }}{lim}\frac{1}{s_n^2}\sum _{k=1}^n{\int }_{|x-{\mu }_k|>\epsilon s_n}{(x-{\mu }_k)}^2𝑑F_k(x)=0.$$

Corollary 1 (Lyapunov’s central limit theorem)

If the Lyapunov condition

$$\frac{1}{s_n^{2+\delta }}\sum _{k=1}^{n}E{|X_k-{\mu }_k|}^{2+\delta }\underset{n\to \mathrm{\infty }}{\overset{}{\to }}0$$

is satisfied for some $\delta >0$, the normal convergence holds.

Corollary 2

If $X_1,X_2,\mathrm{\dots }$ are identically distributed random variables, $E{X}_n=\mu$ and $\mathrm{Var}{S}_n={\sigma }^2$, with , then the normal convergence holds; i.e. $\frac{S_n-n\mu }{\sigma \sqrt{n}}$ converges in distribution (http://planetmath.org/ConvergenceInDistribution) to a random variable with distribution $N(0,1)$.

Reciprocal (Feller)

The reciprocal of Lindeberg’s central limit theorem holds under the following additional assumption:

$$\underset{1\le k\le n}{\max }\left(\frac{{\sigma }_k^2}{s_n^2}\right)\underset{n\to \mathrm{\infty }}{\overset{}{\to }}0.$$

Historical remark

Title	Lindeberg’s central limit theorem
Canonical name	LindebergsCentralLimitTheorem
Date of creation	2013-03-22 13:14:25
Last modified on	2013-03-22 13:14:25
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	19
Author	Koro (127)
Entry type	Theorem
Classification	msc 60F05
Synonym	Lyapunov’s central limit theorem
Synonym	central limit theorem
Synonym	lyapunov condition
Synonym	lindeberg condition
Related topic	TightAndRelativelyCompactMeasures
Defines	normal convergence
Defines	liapunov’s central limit theorem
Defines	liapunov condition