= Weighted central limit theorem with negligible maximum weight
{title2=$\max_k|a_{n,k}|\to0,\quad\sum_ka_{n,k}^2\to v\ \Longrightarrow\ \sum_ka_{n,k}X_k\xrightarrow dN(0,v)$}
Let $X_k$ be <independent and identically distributed random variables> with <expected value> zero, <variance> one, and a <moment-generating function> finite in a neighborhood of zero. Each row of deterministic real weights $a_{n,k}$ is finite. If $\max_k|a_{n,k}|\to0$ and $\sum_ka_{n,k}^2\to v<\infty$, then the weighted sums converge in <distribution> to a <normal distribution> with mean zero and variance $v$; $v=0$ denotes the constant zero.
By the <quadratic expansion of a cumulant-generating function>, write $K(u)=u^2/2+r(u)$ with $r(u)=o(u^2)$. For a fixed real $t$, put $\eta_n=\sup_{0<|u|\leq |t|\max_k|a_{n,k}|}|r(u)|/u^2$, with the supremum taken as zero if its range is empty. Then $\eta_n\to0$ and
$$
\left|\sum_kr(ta_{n,k})\right|\leq\eta_nt^2\sum_ka_{n,k}^2\longrightarrow0.
$$
Thus the log <moment-generating function> of the sum tends to $vt^2/2$. The <convergence theorem for moment-generating functions> proves the conclusion. The supremum is the step that makes the remainder estimate uniform across the row; a separate little-o assertion for each term would not justify summing an increasing number of terms.
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