some facts

inputs/lecture_09.tex

@@ -83,6 +83,35 @@ This will be the weakest notion of convergence, hence it is called
 This notion of convergence will be defined in terms of
 characteristic functions of Fourier transforms.
 
+\subsection{Convolutions${}^\dagger$}
+
+\begin{definition}+[Convolution]
+   Let $\mu$ and $\nu$ be probability measures on $\R^d$
+   with Lebesgue densities $f_\mu$ and $f_\nu$.
+   Then the \vocab{convolution} of $\mu$ and $\nu$,
+   $\mu \ast \nu$,
+   is the probability measure on $\R^d$
+   with Lebesgue density
+   \[
+   f_{\mu \ast \nu}(x) \coloneqq
+       \int_{\R^d} f_\mu(x - t) f\nu(t) \lambda^d(\dif t)
+   \] 
+    
+\end{definition}
+
+\begin{fact}+[Exercise 6.1]
+    If $X_1,X_2,\ldots$ are independent with
+    distributions $X_1 \sim \mu_1$,
+    $X_2 \sim \mu_2, \ldots$,
+    then $X_1 + \ldots + X_n$
+    has distribution
+    \[
+    \mu_1 \ast \mu_2 \ast \ldots \ast \mu_n.
+    \] 
+\end{fact}
+\todo{TODO}
+
+
 \subsection{Characteristic Functions and Fourier Transform}
 
 \begin{definition}
@@ -106,7 +135,30 @@ We have
     \item We have $\phi(0) = 1$.
     \item $|\phi(t)| \le  \int_{\R} |e^{\i t x} | \bP(dx) = 1$.
 \end{itemize}
-\todo{Properties of characteristic functions}
+
+\begin{fact}+
+    Let $X$, $Y$ be independent random variables
+    and $a,b \in \R$.
+    Then
+    \begin{itemize}
+        \item $\phi_{a X + b}(t) = e^{\i t b} \phi_X(\frac{t}{a})$,
+        \item $\phi_{X + Y}(t) = \phi_X(t) + \phi_Y(t)$.
+    \end{itemize}
+\end{fact}
+\begin{proof}
+    We have
+    \begin{IEEEeqnarray*}{rCl}
+        \phi_{a X + b}(t) &=& \bE[e^{\i t (aX + b)}}]\\
+                          &=& e^{\i t b} \bE[e^{\i t a X}]\\
+                          &=& e^{\i t b} \phi_X(\frac{t}{a}).
+    \end{IEEEeqnarray*}
+    Furthermore
+    \begin{IEEEeqnarray*}{rCl}
+        \phi_{X + Y}(t) &=& \bE[e^{\i t (X + Y)}]\\
+                        &=& \bE[e^{\i t X}] \bE[e^{\i t Y}]\\
+                        &=& \phi_X(t) \phi_Y(t).
+    \end{IEEEeqnarray*}
+\end{proof}
 
 \begin{remark}
     Suppose $(\Omega, \cF, \bP)$ is an arbitrary probability space and

inputs/lecture_10.tex

@@ -1,7 +1,5 @@
 \lecture{10}{2023-05-09}{}
 
-% RECAP
-
 First, we will prove some of the most important facts about Fourier transforms.
 
 We consider $(\R, \cB(\R))$.

inputs/lecture_23.tex

@@ -93,7 +93,13 @@ In this lecture we recall the most important point from the lecture.
 
     \item Non-examples: $(\delta_n)_n$
     \item How does one prove weak convergence? How does one write this down in a clear way?
-        % TODO
+        \begin{itemize}
+            \item \autoref{lec10_thm1},
+            \item Levy's continuity theorem
+                \ref{levycontinuity},
+            \item Generalization of Levy's continuity theorem
+                \ref{genlevycontinuity}
+        \end{itemize}
 \end{itemize}
 
 \paragraph{Convolution}
@@ -104,7 +110,6 @@ In this lecture we recall the most important point from the lecture.
 \end{itemize}
 
 
-
 \subsubsubsection{CLT}
 \begin{itemize}
     \item Statement of the CLT