From e566dff88314a4fdabecbac9427087305be50791 Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Wed, 12 Jul 2023 17:50:19 +0200 Subject: [PATCH] typo --- inputs/lecture_08.tex | 2 +- inputs/lecture_12.tex | 2 +- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/inputs/lecture_08.tex b/inputs/lecture_08.tex index a8c2d85..d81c826 100644 --- a/inputs/lecture_08.tex +++ b/inputs/lecture_08.tex @@ -125,7 +125,7 @@ for any $k \in \N$. this also means $\sigma(X_1,X_2,\ldots) \subseteq\sigma\left( \bigcup_{n \in \N} \cF_n \right)$. ``$\subseteq$ '' Since $\cF_n = \sigma(X_1,\ldots,X_n)$, - obviously $\cF_n \subseteq \sigma(X_1,\ldots,X_n)$ + obviously $\cF_n \subseteq \sigma(X_1,X_2\ldots)$ for all $n$. It follows that $\bigcup_{n \in \N} \cF_n \subseteq \sigma(X_1,X_2,\ldots)$. Hence $\sigma\left( \bigcup_{n \in \N} \cF_n \right) \subseteq\sigma(X_1,X_2,\ldots)$. diff --git a/inputs/lecture_12.tex b/inputs/lecture_12.tex index c866f2f..63d95b8 100644 --- a/inputs/lecture_12.tex +++ b/inputs/lecture_12.tex @@ -29,7 +29,7 @@ First, we need to prove some properties of characteristic functions. \begin{refproof}{charfprops} \begin{enumerate}[(i)] \item $\phi_X(0) = \bE[e^{\i 0 X}] = \bE[1] = 1$. - For $t \in \R$, we have $|\phi_X(t)| = |\bE[e^{\i t X}]| \overset{\text{Jensen}}{\le} \bE|e^{\i t X}|] = 1$. + For $t \in \R$, we have $|\phi_X(t)| = |\bE[e^{\i t X}]| \overset{\text{Jensen}}{\le} \bE[|e^{\i t X}|] = 1$. \item Let $t, h \in \R$. Then