= Lindeberg-Feller central limit theorem
{c}
{title2=$\sum_i Z_{n,i}\xrightarrow{d}N(0,\sigma^2)$}
For each $n$, let $Z_{n,i}$ be independent centered <random variables>. If the sum of their <variances> tends to $\sigma^2>0$ and $\sum_i\mathbb E[Z_{n,i}^2\mathbf1_{\{|Z_{n,i}|>\varepsilon\}}]\to0$ for every $\varepsilon>0$, 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>.
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