some small changes

Makefile

@@ -1,5 +1,5 @@
 pdf: init
-	latexmk
+	latexmk < /dev/null
 
 clean:
 	latexmk -c

inputs/a_1_counterexamples.tex

@@ -1,5 +1,17 @@
-\section{Counterexamples}
+\section{(Counter)examples}
+
+Consistent families and inconsistent families
+
+Notions of convergence
 
 Exercise 4.3
 10.2
 
+
+Martingales converging a.s.~but not in $L^1$.
+
+Stopping times
+
+
+
+

inputs/a_2_additional_stuff.tex

@@ -0,0 +1,21 @@
+Important stuff not done in the lecture.
+
+Moments:
+
+$\bE[X^k]$
+
+\begin{lemma}
+    Let $X, Y : \Omega \to [a,b]$
+    If $\bE[X^k] = \bE[Y^k]$,
+    for every $k \in \N_0$
+    then $X = Y$.
+\end{lemma}
+\begin{proof}
+    We have $\bE[p(X)] = \bE[p(Y)]$ for
+    every polynomial $p$.
+    Approximate $e^{\i t X}$
+    with polynomials and use Fourier transforms.
+\end{proof}
+
+
+Laplace transforms

inputs/lecture_02.tex

@@ -1,4 +1,4 @@
-\lecture{2}{}{}
+\lecture{2}{2023-04-11}{Independence, Kolmogorov's consistency theorem, consistent families}
 \section{Independence and Product Measures}
 
 In order to define the notion of independence, we first need to construct
@@ -80,9 +80,9 @@ to an infinite number of random variables.
 
 \begin{theorem}[Kolmogorov extension / consistency theorem]
     \label{thm:kolmogorovconsistency}
-    Informally:
+    \footnote{Informally:
     ``Probability measures are determined by finite-dimensional marginals
-    (as long as these marginals are nice)''
+    (as long as these marginals are nice)''}
 
     Let $\bP_n, n \in \N$ be probability measures on $(\R^n, \cB(\R^n))$
     which are \vocab{consistent},
@@ -117,5 +117,3 @@ to an infinite number of random variables.
     the distribution function of $X_i$.
     In the case of $F_1 = \ldots = F_n$, then $X_1,\ldots, X_n$ are i.i.d.
 \end{example}
-
-

inputs/lecture_03.tex

@@ -154,4 +154,5 @@ we are going to use the following:
         = \lambda_n(C_1) + \lambda_n(C_2)
     \]
     by the definition of the finite product measure.
+    \phantom\qedhere
 \end{refproof}

inputs/lecture_05.tex

@@ -1,4 +1,4 @@
-\lecture{5}{2023-04-21}{}
+\lecture{5}{2023-04-21}{Laws of large numbers}
 \subsection{The Laws of Large Numbers}
 
 
@@ -44,23 +44,14 @@ For the proof of (b) we need the following general result:
 \begin{theorem}
     \label{thm2}
     Let $X_1, X_2, \ldots$ be independent (but not necessarily identically distributed) random variables with $\bE[X_i] = 0$ for all $i$
-    and $\sum_{i=1}^n \Var(X_i) < \infty$.
+    and
+    \[\sum_{i=1}^n \Var(X_i) < \infty.\]
     Then $\sum_{n \ge 1} X_n$ converges almost surely.
 \end{theorem}
+We'll prove this later\todo{Move proof}
 
-\begin{proof}
-
-
-\end{proof}
 \begin{question}
     Does the converse hold? I.e.~does $\sum_{n \ge 1} X_n < \infty$ a.s.~
     then $\sum_{n \ge 1} \Var(X_n) < \infty$.
 \end{question}
 This does not hold. Consider for example $X_n = \frac{1}{n^2} \delta_n + \frac{1}{n^2} \delta_{-n} + (1-\frac{2}{n^2}) \delta_0$.
-
-\begin{refproof}{lln}
-    \begin{enumerate}
-        \item[(b)]
-    \end{enumerate}
-\end{refproof}
-

inputs/lecture_07.tex

@@ -5,6 +5,7 @@
     when the $X_n$ are independent.
 \end{goal}
 \begin{theorem}[Kolmogorov's three-series theorem] % Theorem 3
+    \label{thm:kolmogorovthreeseries}
     \label{thm3}
     Let $X_n$ be a family of independent random variables.
     \begin{enumerate}[(a)]
@@ -43,7 +44,7 @@ For the proof we'll need a slight generalization of \autoref{thm2}:
     almost surely.
     Let $A_n \coloneqq \{\omega : |X_n(\omega)| > C\}$.
     Since the first series $\sum_{n \ge 1}  \bP(A_n) < \infty$,
-    by Borel-Cantelli, $\bP[\text{infinitely many $A_n$ occcur}] = 0$.
+    by Borel-Cantelli, $\bP[\text{infinitely many $A_n$ occur}] = 0$.
 
 
     For the proof of (b), suppose $\sum_{n\ge 1} X_n(\omega) < \infty$

inputs/lecture_09.tex

@@ -80,6 +80,7 @@ This notion of convergence will be defined in terms of characteristic functions
 \subsection{Characteristic Functions and Fourier Transform}
 
 \begin{definition}
+    \label{def:characteristicfunction}
     Consider $(\R, \cB(\R), \bP)$.
     The \vocab{characteristic function} of $\bP$ is defined as
     \begin{IEEEeqnarray*}{rCl}

inputs/lecture_10.tex

@@ -184,6 +184,7 @@ Unfortunately, we won't prove \autoref{bochnersthm} in this lecture.
 
 
 \begin{definition}[Convergence in distribution / weak convergence]
+    \label{def:weakconvergence}
     We say that $\bP_n \subseteq  M_1(\R)$ \vocab[Convergence!weak]{converges weakly} towards $\bP \in M_1(\R)$ (notation: $\bP_n \implies \bP$), iff
     \[
     \forall f \in C_b(\R)~  \int f d\bP_n \to \int f d\bP.

inputs/lecture_13.tex

@@ -264,7 +264,6 @@ which converges.
 $G_1, G_2, \ldots$ is a subsequence of $F_1, F_2,\ldots$.
 However $G_1, G_2, \ldots$ is not converging to $F$,
 as this would fail at $x_0$. This is a contradiction.
-
 \end{refproof}
 
 

inputs/lecture_22.tex

@@ -1,3 +1,4 @@
+\pagebreak
 \lecture{22}{2023-07-04}{Introduction Markov Chains II}
 \section{Markov Chains}
 \todo{Merge this with the end of lecture 21}

inputs/lecture_23.tex

@@ -0,0 +1,170 @@
+\lecture{23}{2023-07-06}{}
+\section{Recap}
+In this lecture we will recall the most important point from the lecture.
+
+\subsection{Construction of iid random variables.}
+
+\begin{itemize}
+    \item Definition of a consistent family (\autoref{def:consistentfamily})
+    \item Important construction:
+
+            Consider a distribution function $F$ and define
+            \[
+                \prod_{i=1}^n (F(b_i) - F(a_i)) \text{\reflectbox{$\coloneqq$}}
+                \mu_n \left( (a_1,b_1] \times x \ldots \times x (a_n, b_n] \right).
+            \]
+
+    \item Examples of consistent and inconsistent families
+        \todo{Exercises}
+    \item Kolmogorov's consistency theorem
+        (\autoref{thm:kolmogorovconsistency})
+\end{itemize}
+
+\subsection{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})
+            \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.
+        \begin{itemize}
+            \item Counter examples showing that $\impliedby$ does not hold in general are important
+            \item $\impliedby$ holds for iid.~uniformly bounded random variables
+            \item Application:
+
+                $\sum_{i=1}^{\infty} \frac{(\pm_1)}{n^{\frac{1}{2} + \epsilon}}$ converges a.s.~for all $\epsilon > 0$.
+
+                $\sum \frac{\pm 1}{ n^{\frac{1}{2} -\epsilon}}$ does not converge a.s.~for any $\epsilon > 0$.
+        \end{itemize}
+    \item Kolmogorov's inequality (\autoref{thm:kolmogorovineq})
+    \item Kolmogorov's $0-1$-law. (\autoref{kolmogorov01})
+
+        In particular, a series of independent random variables converges with probability $0$ or $1$.
+
+    \item Kolmogorov's 3 series theorem. (\autoref{thm:kolmogorovthreeseries})
+        \begin{itemize}
+            \item What are those $3$ series?
+            \item Applications
+        \end{itemize}
+\end{itemize}
+
+\subsubsection{Fourier transform / characteristic functions / weak convegence}
+
+\begin{itemize}
+    \item Definition of Fourier transform
+        (\autoref{def:characteristicfunction})
+    \item The Fourier transform uniquely determines the probability distribution.
+        It is bounded, so many theorems are easily applicable.
+    \item Uniqueness theorem (\autoref{charfuncuniqueness}),
+        inversion formula (\autoref{inversionformula}), ...
+    \item Levy's continuity theorem (\autoref{levycontinuity}),
+        (\autoref{genlevycontinuity})
+    \item Bockner's theorem for positive definite function % TODO REF
+    \item Bockner's theorem for the mass at a point (\autoref{bochnersformula})
+    \item Related notions
+        \todo{TODO}
+        \begin{itemize}
+        \item Laplace transforms $\bE[e^{-\lambda X}]$ for some $\lambda > 0$
+            (not done in the lecture, but still useful).
+        \item Moments $\bE[X^k]$ (not done in the lecture, but still useful)
+            All moments together uniquely determine the distribution.
+    \end{itemize}
+\end{itemize}
+
+\paragraph{Weak convergence}
+\begin{itemize}
+    \item Definition of weak convergence % ( test against continuous, bounded functions).
+        (\autoref{def:weakconvergence})
+    \item Examples:
+        \begin{itemize}
+            \item $(\delta_{\frac{1}{n}})_n$,
+            \item $(\frac{1}{2} \delta_{-\frac{1}{n}} + \frac{1}{2} \delta_{\frac{1}{n}}$,
+            \item $(\cN(0, \frac{1}{n}))_n$,
+            \item $(\frac{1}{n} \delta_n + (1- \frac{1}{n}) \delta_{\frac{1}{n}})_n$.
+        \end{itemize}
+
+    \item Non-examples: $(\delta_n)_n$
+    \item How does one prove weak convergence? How does one write this down in a clear way?
+        % TODO
+\end{itemize}
+
+\paragraph{Convolution}
+\begin{itemize}
+    \item Definition of convolution.
+        \todo{Copy from exercise sheet and write a section 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}
+\begin{itemize}
+    \item Statement of the CLT
+    \item Several versions:
+        \begin{itemize}
+            \item iid (\autoref{clt}),
+            \item Lindeberg (\autoref{lindebergclt}),
+            \item Luyapanov (\autoref{lyapunovclt})
+        \end{itemize}
+    \item How to apply this? Exercises!
+\end{itemize}
+
+\subsection{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 $\bE[ \cdot | \cG]$
+        is the (unique) projection on the closed subspace $L^2(\Omega, \cG, \bP)$.
+        Why is this a closed subspace? Why is the projection orthogonal?
+    \item Radon-Nikodym Theorem (Proof not relevant for the exam)
+    \item (Non-)examples of mutually absolutely continuous measures
+        Singularity in this context? % TODO
+\end{itemize}
+
+\subsection{Martingales}
+
+\begin{itemize}
+    \item Definition of Martingales
+    \item Doob's convergence theorem, Upcrossing inequality
+        (downcrossings for submartingales)
+    \item Examples of Martingales converging a.s.~but not in $L^1$
+    \item Bounded in $L^2$ $\implies$ convergence in $L^2$.
+    \item Martingale increments are orthogonal in $L^2$!
+    \item Doob's (sub-)martingale inequalities
+    \item $\bP[\sup_{k \le n} M_k \ge x]$ $\leadsto$ Look at martingale inequalities!
+        Estimates might come from Doob's inequalities if $(M_k)_k$ is a (sub-)martingale.
+    \item Doob's $L^p$ convergence theorem.
+        \begin{itemize}
+            \item Why is $p > 1$ important? \textbf{Role of Banach-Alaoglu}
+            \item This is an important proof.
+        \end{itemize}
+    \item Uniform integrability % TODO
+    \item What are stopping times?
+    \item (Non-)examples of stopping times
+    \item \textbf{Optional stopping theorem} - be really comfortable with this.
+\end{itemize}
+
+
+\subsection{Markov Chains}
+
+\begin{itemize}
+    \item What are Markov chains?
+    \item State space, initial distribution
+    \item Important examples
+    \item \textbf{What is the relation between Martingales and Markov chains?}
+        $u$ \vocab{harmonic} $\iff Lu = 0$.
+        (sub-/super-) harmonic $u$ $\iff$ for a Markov chain $(X_n)$,
+        $u(X_n)$ is a (sub-/super-)martingale
+    \item Dirichlet problem
+        (Not done in the lecture)
+    \item ... (more in Probability Theory II)
+\end{itemize}

inputs/prerequisites.tex

@@ -3,6 +3,7 @@ from the lecture on stochastic.
 
 \subsection{Notions of Convergence}
 \begin{definition}
+    \label{def:convergence}
     Fix a probability space $(\Omega,\cF,\bP)$.
     Let $X, X_1, X_2,\ldots$ be random variables.
     \begin{itemize}
@@ -30,9 +31,10 @@ from the lecture on stochastic.
             \]
     \end{itemize}
 \end{definition}
-% TODO Connect to ANaIII
+% TODO Connect to AnaIII
 
 \begin{theorem}
+    \label{thm:convergenceimplications}
     \vspace{10pt}
     Let $X$ be a random variable and $X_n, n \in \N$ a sequence of random variables.
     Then
@@ -80,6 +82,9 @@ from the lecture on stochastic.
                        &=& \epsilon \cdot c > 0 \lightning
     \end{IEEEeqnarray*}
     \todo{Improve this with Markov}
+    \todo{Counter examples}
+    \todo{weak convergence}
+    \todo{$L^p$ convergence}
     \end{subproof}
     \begin{claim}
     $X_n \xrightarrow{\bP} X \notimplies X_n\xrightarrow{L^1} X$