 Asymptotic normality of a weighted regression slope

Suppose ${ϵ}_1,{ϵ}_2,\dots$ are independent and identically distributed with mean zero and finite variance ${\sigma }^2>0$. For deterministic weights $x_{n,i}$, put $S_n={\sum }_{i=1}^n{x}_{n,i}^2>0$. Ifthen the Lindeberg-Feller central limit theorem givesThis is the condition that no observation dominates the weighted error sum. In a centered linear regression with orthogonal explanatory columns, ${\hat{\beta }}_1-{\beta }_1={\sum }_i{x}_{n,i}{ϵ}_i/S_n$, 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 ${\delta }_n={\max }_i|a_{n,i}|\to 0$. For each $\eta >0$,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.