# Lindeberg’s central limit theorem

Let $X_{1},X_{2},\dots$ be independent  random variables  with distribution functions  $F_{1},F_{2},\dots$, respectively, such that $EX_{n}=\mu_{n}$ and $\operatorname{Var}X_{n}=\sigma_{n}^{2}<\infty$, with at least one $\sigma_{n}>0$. Let

 $S_{n}=X_{1}+\cdots+X_{n}\;\mbox{and}\;s_{n}=\sqrt{\operatorname{Var}(S_{n})}=% \sqrt{\sigma_{1}^{2}+\cdots+\sigma_{n}^{2}}.$

Then the normalized partial sums $\frac{S_{n}-ES_{n}}{s_{n}}$ 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:

 $\forall\varepsilon>0,\;\lim_{n\rightarrow\infty}\frac{1}{s_{n}^{2}}\sum_{k=1}^% {n}\int_{|x-\mu_{k}|>\varepsilon s_{n}}(x-\mu_{k})^{2}dF_{k}(x)=0.$

Corollary 1 (Lyapunov’s central limit theorem)

If the Lyapunov condition

 $\frac{1}{s_{n}^{2+\delta}}\sum_{k=1}^{n}E|X_{k}-\mu_{k}|^{2+\delta}% \xrightarrow[n\rightarrow\infty]{}0$

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

Corollary 2

If $X_{1},X_{2},\dots$ are identically distributed random variables, $EX_{n}=\mu$ and $\operatorname{Var}S_{n}=\sigma^{2}$, with $0<\sigma<\infty$, 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:

 $\max_{1\leq k\leq n}\left(\frac{\sigma_{k}^{2}}{s_{n}^{2}}\right)\xrightarrow[% n\rightarrow\infty]{}0.$

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