Lyapunov CLT

Like the Lindeberg-Feller CLT, the Lyapunov CLT is for independent observations without a common mean or variance.

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

$$\underset{n}{\lim }\frac{1}{s_n^3}\sum _{i\le n}\mathbb{E}|X_i-{\mu }_i{|}^3=0,$$

then

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

The Lyapunov CLT replaces the Lindeberg condition with a Lyapunov condition, which is a condition of the third moments. It also recovers the usual CLT for iid random variables with finite variance.

The Lyapunov result is weaker than the Lindeberg-Feller CLT, as the Lyapunov condition implies the Lindeberg condition.