 Lindeberg-Feller central limit theorem

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.