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Josia Pietsch 2023-07-13 00:38:46 +02:00
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10
inputs/lecture_10.tex
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@ -123,7 +123,7 @@ However, Fourier analysis is not only useful for continuous probability density
Let $\bP \in M_1(\lambda)$. Let $\bP \in M_1(\lambda)$.
Then Then
\[ \[
\forall x \in \R ~ \bP\left( \{x\} \right) = \lim_{T \to \infty} \frac{1}{2 T} \int_{-T}^T e^{-\i t x } \phi(t) \dif t. \forall x \in \R .~ \bP\left( \{x\} \right) = \lim_{T \to \infty} \frac{1}{2 T} \int_{-T}^T e^{-\i t x } \phi(t) \dif t.
\] \]
\end{theorem} \end{theorem}
@ -175,18 +175,18 @@ However, Fourier analysis is not only useful for continuous probability density
&=& \int_{\R} \left| \sum_{l} c_l e^{\i t_l x}\right|^2 \ge 0 &=& \int_{\R} \left| \sum_{l} c_l e^{\i t_l x}\right|^2 \ge 0
\end{IEEEeqnarray*} \end{IEEEeqnarray*}
\end{refproof} \end{refproof}
\begin{theorem}[Bochner's theorem]\label{bochnersthm} \begin{theorem}[Bochner's theorem]\label{thm:bochner}
The converse to \autoref{thm:lec_10thm5} holds, i.e.~any The converse to \autoref{thm:lec_10thm5} holds, i.e.~any
$\phi: \R \to \C$ satisfying (a) and (b) of \autoref{thm:lec_10thm5} $\phi: \R \to \C$ satisfying (a) and (b) of \autoref{thm:lec_10thm5}
must be the Fourier transform of a probability measure $\bP$ must be the Fourier transform of a probability measure $\bP$
on $(\R, \cB(\R))$. on $(\R, \cB(\R))$.
\end{theorem} \end{theorem}
Unfortunately, we won't prove \autoref{bochnersthm} in this lecture. Unfortunately, we won't prove \autoref{thm:bochner} in this lecture.
\begin{definition}[Convergence in distribution / weak convergence] \begin{definition}[Convergence in distribution / weak convergence]
\label{def:weakconvergence} \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 We say that $\bP_n \in M_1(\R)$ \vocab[Convergence!weak]{converges weakly} towards $\bP \in M_1(\R)$ (notation: $\bP_n \implies \bP$), iff
\[ \[
\forall f \in C_b(\R)~ \int f \dif\bP_n \to \int f \dif\bP. \forall f \in C_b(\R)~ \int f \dif\bP_n \to \int f \dif\bP.
\] \]
@ -245,6 +245,8 @@ Unfortunately, we won't prove \autoref{bochnersthm} in this lecture.
$\bP_n \implies \bP$, where $\bP_n$ is the distribution of $X_n$ $\bP_n \implies \bP$, where $\bP_n$ is the distribution of $X_n$
and $\bP$ is the distribution of $X$. and $\bP$ is the distribution of $X$.
\end{definition} \end{definition}
It is easy to see, that this is equivalent to $\bE[f(X_n)] \to \bE[f(X)]$
for all $f \in C_b(\R)$.
\begin{example} \begin{example}
Let $X_n \coloneqq \frac{1}{n}$ Let $X_n \coloneqq \frac{1}{n}$
and $F_n$ the distribution function, i.e.~$F_n = \One_{[\frac{1}{n},\infty)}$. and $F_n$ the distribution function, i.e.~$F_n = \One_{[\frac{1}{n},\infty)}$.

4
inputs/lecture_23.tex
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@ -67,8 +67,8 @@ In this lecture we recall the most important point from the lecture.
inversion formula (\autoref{inversionformula}), ... inversion formula (\autoref{inversionformula}), ...
\item Levy's continuity theorem (\autoref{levycontinuity}), \item Levy's continuity theorem (\autoref{levycontinuity}),
(\autoref{genlevycontinuity}) (\autoref{genlevycontinuity})
\item Bockner's theorem for positive definite function % TODO REF \item Bochner's theorem for positive definite function (\autoref{thm:bochner})
\item Bockner's theorem for the mass at a point (\autoref{bochnersformula}) \item Bochner's theorem for the mass at a point (\autoref{bochnersformula})
\item Related notions \item Related notions
\todo{TODO} \todo{TODO}
\begin{itemize} \begin{itemize}