= Asymptotic normality of a weighted regression slope
{title2=$\max_i x_{n,i}^2/\sum_i x_{n,i}^2\to0$}
Suppose $\epsilon_1,\epsilon_2,\ldots$ are <independent and identically distributed> with <mean> zero and finite <variance> $\sigma^2>0$. For deterministic weights $x_{n,i}$, put $S_n=\sum_{i=1}^n x_{n,i}^2>0$. If
$$
\frac{\max_i x_{n,i}^2}{S_n}\longrightarrow0,
$$
then the <Lindeberg-Feller central limit theorem> gives
$$
\frac{\sum_i x_{n,i}\epsilon_i}{\sigma\sqrt{S_n}}\xrightarrow d N(0,1).
$$
This is the condition that no observation dominates the weighted error sum. In a centered <linear regression> with <orthogonal> explanatory columns, $\widehat\beta_1-\beta_1=\sum_i x_{n,i}\epsilon_i/S_n$, so the result justifies a <normal approximation> with <variance> $\sigma^2/S_n$.
To verify the <Lindeberg condition>, write $a_{n,i}=x_{n,i}/\sqrt{S_n}$ and $\delta_n=\max_i|a_{n,i}|\to0$. For each $\eta>0$,
$$
\sum_i\mathbb E\left[\left(\frac{a_{n,i}\epsilon_i}{\sigma}\right)^2\mathbf1_{\{|a_{n,i}\epsilon_i|>\eta\sigma\}}\right]
\le\frac1{\sigma^2}\mathbb E\left[\epsilon_1^2\mathbf1_{\{|\epsilon_1|>\eta\sigma/\delta_n\}}\right]\longrightarrow0.
$$
The inequality uses $\sum_i a_{n,i}^2=1$, and the limit follows from the finite second moment. Without the maximum-weight condition, adding observations with zero explanatory value can leave the slope determined by a fixed number of nonnormal errors for every sample size.
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