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fixed Lindeberg CLT

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Josia Pietsch 2023-07-12 23:52:51 +02:00
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inputs/lecture_13.tex
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@ -16,7 +16,9 @@ if $X_1, X_2,\ldots$ are i.i.d.~with $ \mu = \bE[X_1]$,
Assume $X_1, X_2, \ldots,$ are independent (but not necessarily identically distributed) with $\mu_i = \bE[X_i] < \infty$ and $\sigma_i^2 = \Var(X_i) < \infty$. Assume $X_1, X_2, \ldots,$ are independent (but not necessarily identically distributed) with $\mu_i = \bE[X_i] < \infty$ and $\sigma_i^2 = \Var(X_i) < \infty$.
Let $S_n = \sqrt{\sum_{i=1}^{n} \sigma_i^2}$ Let $S_n = \sqrt{\sum_{i=1}^{n} \sigma_i^2}$
and assume that and assume that
\[\lim_{n \to \infty} \frac{1}{S_n^2} \bE\left[(X_i - \mu_i)^2 \One_{|X_i - \mu_i| > \epsilon S_n}\right] = 0\] \[\lim_{n \to \infty} \frac{1}{S_n^2} \sum_{i=1}^{n} \bE\left[
(X_i - \mu_i)^2 \One_{|X_i - \mu_i| > \epsilon S_n}
\right] = 0\]
for all $\epsilon > 0$ for all $\epsilon > 0$
(\vocab{Lindeberg condition}\footnote{``The truncated variance is negligible compared to the variance.''}). (\vocab{Lindeberg condition}\footnote{``The truncated variance is negligible compared to the variance.''}).