Lindeberg-Feller central limit theorem (${\sum }_i{Z}_{n,i}\stackrel{d}{\to }N(0,{\sigma }^2)$)

For each $n$, let $Z_{n,i}$ be independent centered random variables. If the sum of their variances tends to ${\sigma }^2>0$ and ${\sum }_i\mathbb{E}[Z_{n,i}^2{1}_{\{|Z_{n,i}|>\epsilon \}}]\to 0$ for every $\epsilon >0$, their sum converges in distribution to the centered normal distribution with variance ${\sigma }^2$. The second hypothesis is the Lindeberg condition. If the total variance tends to zero, the corresponding degenerate limit follows directly from the Chebyshev inequality.

For independent centered random variables with total variance $s_n^2$, the Lyapunov condition requires the displayed expression to tend to zero for some $\delta >0$. The inequality $z^2{1}_{\{|z|>\epsilon s_n\}}\le |z{|}^{2+\delta }/(\epsilon s_n{)}^{\delta }$ implies the Lindeberg condition after normalization. The Lindeberg-Feller central limit theorem then makes the normalized sum converge to the standard normal distribution.