summary

inputs/lecture_16.tex

@@ -74,6 +74,16 @@ The Radon Nikodym theorem is the converse of that:
     w.r.t.~$\mu$.
     This is written as $\nu \ll \mu$.
 \end{definition}
+\begin{definition}+
+    Two measures $\mu$ and $\nu$ on a measure space $(\Omega, \cF)$
+    are called \vocab{singular},
+    denoted $\mu \bot \nu$,
+    iff there exists $A \in \cF$ such that
+    \[
+    \mu(A) = \nu(A^c) = 0.
+    \]
+\end{definition}
+
 
 With \autoref{radonnikodym} we get a very short proof of the existence
 of conditional expectation:
@@ -171,6 +181,7 @@ we have gathered up to time $n$.
 Typically $\cF_n = \sigma(X_1, \ldots, X_n)$ for a sequence of random variables.
 
 \begin{definition}
+    \label{def:martingale}
     Let $(\cF_n)$ be a filtration and
     $X_1,\ldots,X_n$ be random variables such that $X_i \in L^1(\bP)$.
     Then we say that $(X_n)_{n \ge 1}$ is an $(\cF_n)_n$-\vocab{martingale}

inputs/lecture_18.tex

@@ -20,6 +20,7 @@ Hence the same holds for submartingales, i.e.
 \end{question}
 
 \begin{example}[A martingale not converging in $L^1$ ]
+    \label{ex:martingale-not-converging-in-l1}
     Fix $u > 1$ and let $p = \frac{1}{1+u}$.
     Let $ (Z_n)_{n \ge 1}$ be i.i.d.~$\pm 1$ with
     $\bP[Z_n = 1] = p$.

inputs/lecture_19.tex

@@ -23,6 +23,7 @@
 \end{goal}
 
 \begin{definition}
+    \label{def:ui}
     A sequence of random variables $(X_n)_n$ is called \vocab{uniformly integrable} (UI),
     if
     \[\forall  \epsilon > 0 .~\exists  K > 0 .~ \forall n.~
@@ -245,4 +246,5 @@ Let $(\Omega, \cF, \bP)$ as always and let $(\cF_n)_n$ always be a filtration.
     Let $(X_n)_n$ be a martingale bounded in $L^p$.
     Then there exists a random variable $X \in  L^p$, such that
     $X_n = \bE[X | \cF_n]$ for all $n$.
+    In particular, $X_n \xrightarrow{L^p} X$.
 \end{theorem}

inputs/lecture_23.tex

@@ -117,7 +117,7 @@ In this lecture we recall the most important point from the lecture.
         \begin{itemize}
             \item iid (\autoref{clt}),
             \item Lindeberg (\autoref{lindebergclt}),
-            \item Luyapanov (\autoref{lyapunovclt})
+            \item Lyapanov (\autoref{lyapunovclt})
         \end{itemize}
     \item How to apply this? Exercises!
 \end{itemize}
@@ -125,10 +125,12 @@ In this lecture we recall the most important point from the lecture.
 \subsubsection{Conditional expectation}
 \begin{itemize}
     \item Definition and existence of conditional expectation for $X \in L^1(\Omega, \cF, \bP)$
+        (\autoref{conditionalexpectation})
     \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 Radon-Nikodym Theorem \ref{radonnikodym}
+        (Proof not relevant for the exam)
     \item (Non-)examples of mutually absolutely continuous measures
         Singularity in this context? % TODO
 \end{itemize}
@@ -136,25 +138,31 @@ In this lecture we recall the most important point from the lecture.
 \subsubsection{Martingales}
 
 \begin{itemize}
-    \item Definition of Martingales
+    \item Definition of Martingales (\autoref{def:martingale})
-    \item Doob's convergence theorem, Upcrossing inequality
+    \item Doob's convergence theorem (\autoref{doobmartingaleconvergence}),
+        Upcrossing inequality (\autoref{lec17l1}, \autoref{lec17l2}, \autoref{lec17l3})
         (downcrossings for submartingales)
     \item Examples of Martingales converging a.s.~but not in $L^1$
-    \item Bounded in $L^2$ $\implies$ convergence in $L^2$.
+        (\autoref{ex:martingale-not-converging-in-l1})
+    \item Bounded in $L^2$ $\implies$ convergence in $L^2$
+        (\autoref{martingaleconvergencel2}).
     \item Martingale increments are orthogonal in $L^2$!
+        (\autoref{martingaleincrementsorthogonal})
     \item Doob's (sub-)martingale inequalities
+        (\autoref{dooblp}),
     \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.
+    \item Doob's $L^p$ convergence theorem
+        (\autoref{ceismartingale}, \autoref{martingaleisce}).
         \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 Uniform integrability (\autoref{def:ui})
-    \item What are stopping times? \autoref{def:stopping-time}
+    \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.
+    \item \textbf{Optional stopping theorem} (\autoref{optionalstopping})
-        \autoref{optionalstopping}
+        - be really comfortable with this.
 \end{itemize}