typo

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 $\sigma\left( \bigcup_{n \in \N} \cF_n \right) \subseteq\sigma(X_1,X_2,\ldots)$.

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