some facts

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Josia Pietsch 2023-07-18 15:57:08 +02:00
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@ -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 This notion of convergence will be defined in terms of
characteristic functions of Fourier transforms. 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} \subsection{Characteristic Functions and Fourier Transform}
\begin{definition} \begin{definition}
@ -106,7 +135,30 @@ We have
\item We have $\phi(0) = 1$. \item We have $\phi(0) = 1$.
\item $|\phi(t)| \le \int_{\R} |e^{\i t x} | \bP(dx) = 1$. \item $|\phi(t)| \le \int_{\R} |e^{\i t x} | \bP(dx) = 1$.
\end{itemize} \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} \begin{remark}
Suppose $(\Omega, \cF, \bP)$ is an arbitrary probability space and Suppose $(\Omega, \cF, \bP)$ is an arbitrary probability space and

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

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