boundedness

inputs/a_2_additional_stuff.tex

@@ -1,21 +1,66 @@
-Important stuff not done in the lecture.
+\section{Additional Material}
 
-Moments:
+% Important stuff not done in the lecture.
 
-$\bE[X^k]$
+\subsection{Notions of boundedness}
+The following is just a short overview of all the notions of
+boundedness we used in the lecture.
 
-\begin{lemma}
+\begin{definition}+[Boundedness]
-    Let $X, Y : \Omega \to [a,b]$
+    Let $\cX$ be a set of random variables.
-    If $\bE[X^k] = \bE[Y^k]$,
+    We say that $\cX$ is
-    for every $k \in \N_0$
+    \begin{itemize}
-    then $X = Y$.
+        \item \vocab{uniformly bounded} iff
-\end{lemma}
+            \[\sup_{X \in \cX} \sup_{\omega \in \Omega} |X(\omega)| < \infty,\]
-\begin{proof}
+        \item \vocab{dominated by $f \in L^p$} for $p \ge 1$ iff
-    We have $\bE[p(X)] = \bE[p(Y)]$ for
+            \[
-    every polynomial $p$.
+            \forall X \in \cX .~ |X| \le f,
-    Approximate $e^{\i t X}$
+            \] 
-    with polynomials and use Fourier transforms.
+        \item \vocab{bounded in $L^p$} for $p \ge 1$ iff
-\end{proof}
+            \[
+            \sup_{X \in \cX} \|X\|_{L^p} < \infty,
+            \] 
+        \item \vocab{uniformly integrable} iff
+            \[
+            \forall \epsilon > 0 .~\exists K .~ \forall X \in \cX.~
+            \bE[|X| \One_{|X| > K}] < \epsilon.
+            \] 
+    \end{itemize}
+    
+\end{definition}
+
+\begin{fact}+
+    Let $\cX$ be a set of random variables.
+    Let $1 < p \le q < \infty$
+    Then the following implications hold:
+    \begin{figure}[H]
+        \centering
+        \begin{tikzpicture}
+            \node at (0,2.5) (ub) {$\cX$ is uniformly bounded};
+            \node at (-2.5,1.5) (dq) {$\cX$ is dominated by $f \in L^q$};
+            \node at (-2.5,0.5) (dp) {$\cX$ is dominated by $f \in L^p$};
+            \node at (2.5,1.0) (bq) {$\cX$ is bounded in $L^q$};
+            \node at (2.5,0) (bp) {$\cX$ is bounded in $L^p$};
+            \node at (-2.5,-0.5) (d1) {$\cX$ is dominated by $f \in L^1$};
+            \node at (0,-1.5) (ui) {$\cX$ is uniformly integrable};
+            \node at (2.5,-2.5) (b1) {$\cX$ is bounded in $L^1$};
+            \draw[double equal sign distance, -implies] (ub) -- (dq);
+            % \draw[double equal sign distance, -implies] (ub) -- (bq);
+            \draw[double equal sign distance, -implies] (bq) -- (bp);
+            \draw[double equal sign distance, -implies] (dq) -- (dp);
+            \draw[double equal sign distance, -implies] (dq) -- (bq);
+            \draw[double equal sign distance, -implies] (dp) -- (bp);
+            \draw[double equal sign distance, -implies] (bp) -- (ui);
+            \draw[double equal sign distance, -implies] (dp) -- (d1);
+            \draw[double equal sign distance, -implies] (d1) -- (ui);
+            \draw[double equal sign distance, -implies] (ui) -- (b1);
+        \end{tikzpicture}
+    \end{figure}
+\end{fact}
 
 
-Laplace transforms
+
+
+
+\subsection{Laplace Transforms}
+\todo{Write something about Laplace Transforms}

inputs/lecture_01.tex

@@ -22,7 +22,11 @@ First, let us recall some basic definitions:
             \end{itemize}
     \end{itemize}
 \end{definition}
-
+\begin{definition}+
+    Let $X$ be a random variable and $k \in \N$.
+    Then the $k$-th \vocab{moment} of $X$ is defined as
+    $\bE[X^k]$.
+\end{definition}
 
 \begin{definition}
     A \vocab{random variable} $X : (\Omega, \cF) \to (\R, \cB(\R))$
@@ -63,6 +67,7 @@ The converse to this fact is also true:
     See theorem 2.4.3 in Stochastik.
 \end{proof}
 
+
 \begin{example}[Some important probability distribution functions]\hfill
     \begin{enumerate}[(1)]
         \item \vocab{Uniform distribution} on $[0,1]$:

inputs/lecture_07.tex

@@ -161,7 +161,7 @@ More formally:
     The idea of `` $\implies$ '' will lead to coupling. % TODO ?
 \end{remark}
 A proof of \autoref{thm5} can be found in the notes.\notes
-\begin{example}[Application of \autoref{thm5}]
+\begin{example}[Application of \autoref{thm4}]
     The series $\sum_{n}  \frac{1}{n^{\frac{1}{2} + \epsilon}}$
     does not converge for $\epsilon < \frac{1}{2}$.
     However

inputs/lecture_19.tex

@@ -183,7 +183,7 @@ However, some subsets can be easily described, e.g.
 
     Since $\phi$ is Lipschitz,
     $ X_n \xrightarrow{\bP} X \implies \phi(X_n) \xrightarrow{\bP} \phi(X)$.
-    By the bounded convergence theorem % TODO
+    By the bounded convergence theorem, \autoref{thm:boundedconvergence},
     $|\phi(X_n)| \le k \implies \int |  \phi(X_n) - \phi(X)| \dif \bP \to  0$.
 
 

inputs/lecture_20.tex

@@ -122,6 +122,7 @@ we need the following theorem, which we won't prove here:
 \subsection{Stopping Times}
 
 \begin{definition}[Stopping time]
+    \label{def:stopping-time}
     A random variable $T: \Omega \to \N_0 \cup  \{\infty\}$ on a filtered probability space $(\Omega, \cF, \{\cF_n\}_n, \bP)$ is called a \vocab{stopping time},
     if
     \[

inputs/lecture_23.tex

@@ -151,9 +151,10 @@ In this lecture we recall the most important point from the lecture.
             \item This is an important proof.
         \end{itemize}
     \item Uniform integrability % TODO
-    \item What are stopping times?
+    \item What are stopping times? \autoref{def:stopping-time}
     \item (Non-)examples of stopping times
     \item \textbf{Optional stopping theorem} - be really comfortable with this.
+        \autoref{optionalstopping}
 \end{itemize}
 
 

probability_theory.tex

@@ -17,7 +17,7 @@
 \tableofcontents
 \cleardoublepage
 
-%\mainmatter
+%\mainatter
 
 \input{inputs/intro.tex}
 
@@ -55,6 +55,8 @@
 \section{Appendix}
 \input{inputs/a_0_distributions.tex}
 \end{landscape}
+\pagebreak
+\input{inputs/a_2_additional_stuff.tex}
 \input{inputs/lecture_23.tex}
 
 \cleardoublepage