From ecf530d05f597a09504beaf210e3e2f5b2e0f6b3 Mon Sep 17 00:00:00 2001
From: Josia Pietsch <josia.pietsch@uni-bonn.de>
Date: Thu, 13 Jul 2023 00:17:26 +0200
Subject: [PATCH] Lyapunov CLT

---
 inputs/lecture_13.tex | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/inputs/lecture_13.tex b/inputs/lecture_13.tex
index ae3b09f..87e5e66 100644
--- a/inputs/lecture_13.tex
+++ b/inputs/lecture_13.tex
@@ -35,7 +35,8 @@ if $X_1, X_2,\ldots$ are i.i.d.~with $ \mu = \bE[X_1]$,
     and $S_n \coloneqq  \sqrt{\sum_{i=1}^n \sigma_i^2}$.
     Then, assume that, for some $\delta > 0$,
     \[
-    \lim_{n \to \infty} \sum_{i=1}^{n} \bE[(X_i - \mu_i)^{2 + \delta}] = 0
+    \lim_{n \to \infty} \frac{1}{S_n^{2+\delta}}
+    \sum_{i=1}^{n} \bE[(X_i - \mu_i)^{2 + \delta}] = 0
     \]
     (\vocab{Lyapunov condition}).
     Then the CLT holds.