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 $\frac{S_n-E{S}_n}{s_n}$ convergein 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