some small changes

inputs/lecture_23.tex

@@ -1,8 +1,8 @@
-\lecture{23}{2023-07-06}{}
-\section{Recap}
-In this lecture we will recall the most important point from the lecture.
+\lecture{23}{2023-07-06}{Recap}
+\subsection{Recap}
+In this lecture we recall the most important point from the lecture.
 
-\subsection{Construction of iid random variables.}
+\subsubsection{Construction of iid random variables.}
 
 \begin{itemize}
     \item Definition of a consistent family (\autoref{def:consistentfamily})
@@ -20,20 +20,18 @@ In this lecture we will recall the most important point from the lecture.
         (\autoref{thm:kolmogorovconsistency})
 \end{itemize}
 
-\subsection{Limit theorems}
+\subsubsection{Limit theorems}
 \begin{itemize}
     \item Work with iid.~random variables.
     \item Notions of convergence (\autoref{def:convergence})
     \item Implications between different notions of convergence (very important) and counter examples.
         (\autoref{thm:convergenceimplications})
 
-    \item \begin{itemize}
-        \item Laws of large numbers: (\autoref{lln})
+    \item \item Laws of large numbers: (\autoref{lln})
             \begin{itemize}
                 \item WLLN: convergence in probability
                 \item SLLN: weak convergence
             \end{itemize}
-    \end{itemize}
     \item \autoref{thm2} (building block for SLLN):
         Let $(X_n)$ be independent with mean $0$ and $\sum \sigma_n^2 < \infty$,
         then $ \sum X_n $ converges a.s.
@@ -58,7 +56,7 @@ In this lecture we will recall the most important point from the lecture.
         \end{itemize}
 \end{itemize}
 
-\subsubsection{Fourier transform / characteristic functions / weak convegence}
+\subsubsubsection{Fourier transform / characteristic functions / weak convegence}
 
 \begin{itemize}
     \item Definition of Fourier transform
@@ -101,13 +99,13 @@ In this lecture we will recall the most important point from the lecture.
 \paragraph{Convolution}
 \begin{itemize}
     \item Definition of convolution.
-        \todo{Copy from exercise sheet and write a section about this}
+        \todo{Copy from exercise sheet and write a subsection about this}
     \item $X_i \sim \mu_i \text{ iid. }\implies X_1 + \ldots + X_n \sim \mu_1 \ast \ldots \ast \mu_n$.
 \end{itemize}
 
 
 
-\subsubsection{CLT}
+\subsubsubsection{CLT}
 \begin{itemize}
     \item Statement of the CLT
     \item Several versions:
@@ -119,7 +117,7 @@ In this lecture we will recall the most important point from the lecture.
     \item How to apply this? Exercises!
 \end{itemize}
 
-\subsection{Conditional expectation}
+\subsubsection{Conditional expectation}
 \begin{itemize}
     \item Definition and existence of conditional expectation for $X \in L^1(\Omega, \cF, \bP)$
     \item If $H = L^2(\Omega, \cF, \bP)$, then $\bE[ \cdot | \cG]$
@@ -130,7 +128,7 @@ In this lecture we will recall the most important point from the lecture.
         Singularity in this context? % TODO
 \end{itemize}
 
-\subsection{Martingales}
+\subsubsection{Martingales}
 
 \begin{itemize}
     \item Definition of Martingales
@@ -154,7 +152,7 @@ In this lecture we will recall the most important point from the lecture.
 \end{itemize}
 
 
-\subsection{Markov Chains}
+\subsubsection{Markov Chains}
 
 \begin{itemize}
     \item What are Markov chains?

inputs/prerequisites.tex

@@ -73,7 +73,9 @@ from the lecture on stochastic.
     \begin{subproof}
     We have $\bE[|X_n - X|] \to  0$.
     Suppose there exists an $\epsilon > 0$ such that
-    $\lim_{n \to \infty} \bP[|X_n - X| > \epsilon] = c > 0$.
+    $\limsup\limits_{n \to \infty} \bP[|X_n - X| > \epsilon] = c > 0$.
+    W.l.o.g.~$\lim_{n \to \infty} \bP[|X_n - X| > \epsilon] = c$,
+    otherwise choose an appropriate subsequence.
     We have
     \begin{IEEEeqnarray*}{rCl}
         \bE[|X_n - X|] &=& \int_\Omega |X_n - X | d\bP\\
@@ -90,7 +92,7 @@ from the lecture on stochastic.
     $X_n \xrightarrow{\bP} X \notimplies X_n\xrightarrow{L^1} X$
     \end{claim}
     \begin{subproof}
-    Take $([0,1], \cB([0,1 ]), \lambda)([0,1], \cB([0,1 ]), \lambda)$
+    Take $([0,1], \cB([0,1 ]), \lambda)$
     and define $X_n \coloneqq  n \One_{[0, \frac{1}{n}]}$.
     We have $\bP[|X_n| > \epsilon] = \frac{1}{n}$
     for $n$ large enough.