From ab1a893dfc0bd116e8f6c7989c52db80653da540 Mon Sep 17 00:00:00 2001
From: Josia Pietsch <josia.pietsch@uni-bonn.de>
Date: Wed, 12 Jul 2023 23:52:51 +0200
Subject: [PATCH] fixed Lindeberg CLT

---
 inputs/lecture_13.tex | 4 +++-
 1 file changed, 3 insertions(+), 1 deletion(-)

diff --git a/inputs/lecture_13.tex b/inputs/lecture_13.tex
index aea6d1a..ae3b09f 100644
--- a/inputs/lecture_13.tex
+++ b/inputs/lecture_13.tex
@@ -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 $\sigma_i^2 = \Var(X_i) < \infty$.
     Let $S_n = \sqrt{\sum_{i=1}^{n} \sigma_i^2}$
     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 $\epsilon > 0$
     (\vocab{Lindeberg condition}\footnote{``The truncated variance is negligible compared to the variance.''}).