Lindeberg-Feller CLT

A CLT for independent random variables with finite variances.

Let $X_1,X_2,\dots$ be independent with $\mathbb{E}[X_i]={\mu }_i$ and $\mathbb{V}(X_i)={\sigma }_i^2$. Let $s_n^2={\sum }_{i\le n}{\sigma }_i^2$ and

$$L_n(ϵ)=\frac{1}{s_n^2}\sum _{i\le n}\mathbb{E}[|X_i-{\mu }_i^2|1(|X_i-{\mu }_i|>ϵs_n)].$$

If $L_n(ϵ)\to 0$ as $n\to \infty$ for all fixed $ϵ>0$, then we have the following CLT:

$$\frac{1}{s_n}\sum _{i\le n}(X_i-{\mu }_i)\stackrel{d}{\to }N(0,1).$$

Note that this implies the “usual” CLT for iid random variables with finite variance ${\sigma }^2$, since in that case $L_n(ϵ)=\mathbb{E}[|X-\mu {|}^{2}1(|X-\mu |>ϵ\sigma \sqrt{n})]/{\sigma }^2$ which certainly goes to 0 as $n\to \infty$. In that case we get the CLT

$$\sqrt{n}\left(\frac{1}{n}\sum _{i\le n}{X}_i-\mu \right)\to N(0,{\sigma }^2).$$

The quantity $L_n(ϵ)$ is called the Lindeberg function, and $L_n(ϵ)\to 0$ is called the Lindeberg condition.

The Lindeberg-Feller CLT strengthens the Lyapunov CLT.