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Josia Pietsch 2023-07-12 17:50:19 +02:00
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inputs/lecture_08.tex
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@ -125,7 +125,7 @@ for any $k \in \N$.
this also means $\sigma(X_1,X_2,\ldots) \subseteq\sigma\left( \bigcup_{n \in \N} \cF_n \right)$. this also means $\sigma(X_1,X_2,\ldots) \subseteq\sigma\left( \bigcup_{n \in \N} \cF_n \right)$.
``$\subseteq$ '' Since $\cF_n = \sigma(X_1,\ldots,X_n)$, ``$\subseteq$ '' 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 $n$. for all $n$.
It follows that $\bigcup_{n \in \N} \cF_n \subseteq \sigma(X_1,X_2,\ldots)$. It follows that $\bigcup_{n \in \N} \cF_n \subseteq \sigma(X_1,X_2,\ldots)$.
Hence $\sigma\left( \bigcup_{n \in \N} \cF_n \right) \subseteq\sigma(X_1,X_2,\ldots)$. Hence $\sigma\left( \bigcup_{n \in \N} \cF_n \right) \subseteq\sigma(X_1,X_2,\ldots)$.

2
inputs/lecture_12.tex
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@ -29,7 +29,7 @@ First, we need to prove some properties of characteristic functions.
\begin{refproof}{charfprops} \begin{refproof}{charfprops}
\begin{enumerate}[(i)] \begin{enumerate}[(i)]
\item $\phi_X(0) = \bE[e^{\i 0 X}] = \bE[1] = 1$. \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 $t, h \in \R$. \item Let $t, h \in \R$.
Then Then