some small changes
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@ -123,7 +123,7 @@ However, Fourier analysis is not only useful for continuous probability density
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Let $\bP \in M_1(\lambda)$.
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Let $\bP \in M_1(\lambda)$.
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Then
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Then
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\[
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\[
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\forall x \in \R ~ \bP\left( \{x\} \right) = \lim_{T \to \infty} \frac{1}{2 T} \int_{-T}^T e^{-\i t x } \phi(t) \dif t.
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\forall x \in \R .~ \bP\left( \{x\} \right) = \lim_{T \to \infty} \frac{1}{2 T} \int_{-T}^T e^{-\i t x } \phi(t) \dif t.
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\]
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\]
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\end{theorem}
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\end{theorem}
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@ -175,18 +175,18 @@ However, Fourier analysis is not only useful for continuous probability density
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&=& \int_{\R} \left| \sum_{l} c_l e^{\i t_l x}\right|^2 \ge 0
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&=& \int_{\R} \left| \sum_{l} c_l e^{\i t_l x}\right|^2 \ge 0
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\end{IEEEeqnarray*}
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\end{IEEEeqnarray*}
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\end{refproof}
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\end{refproof}
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\begin{theorem}[Bochner's theorem]\label{bochnersthm}
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\begin{theorem}[Bochner's theorem]\label{thm:bochner}
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The converse to \autoref{thm:lec_10thm5} holds, i.e.~any
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The converse to \autoref{thm:lec_10thm5} holds, i.e.~any
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$\phi: \R \to \C$ satisfying (a) and (b) of \autoref{thm:lec_10thm5}
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$\phi: \R \to \C$ satisfying (a) and (b) of \autoref{thm:lec_10thm5}
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must be the Fourier transform of a probability measure $\bP$
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must be the Fourier transform of a probability measure $\bP$
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on $(\R, \cB(\R))$.
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on $(\R, \cB(\R))$.
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\end{theorem}
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\end{theorem}
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Unfortunately, we won't prove \autoref{bochnersthm} in this lecture.
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Unfortunately, we won't prove \autoref{thm:bochner} in this lecture.
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\begin{definition}[Convergence in distribution / weak convergence]
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\begin{definition}[Convergence in distribution / weak convergence]
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\label{def:weakconvergence}
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\label{def:weakconvergence}
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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
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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
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\[
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\[
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\forall f \in C_b(\R)~ \int f \dif\bP_n \to \int f \dif\bP.
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\forall f \in C_b(\R)~ \int f \dif\bP_n \to \int f \dif\bP.
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\]
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\]
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@ -245,6 +245,8 @@ Unfortunately, we won't prove \autoref{bochnersthm} in this lecture.
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$\bP_n \implies \bP$, where $\bP_n$ is the distribution of $X_n$
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$\bP_n \implies \bP$, where $\bP_n$ is the distribution of $X_n$
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and $\bP$ is the distribution of $X$.
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and $\bP$ is the distribution of $X$.
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\end{definition}
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\end{definition}
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It is easy to see, that this is equivalent to $\bE[f(X_n)] \to \bE[f(X)]$
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for all $f \in C_b(\R)$.
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\begin{example}
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\begin{example}
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Let $X_n \coloneqq \frac{1}{n}$
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Let $X_n \coloneqq \frac{1}{n}$
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and $F_n$ the distribution function, i.e.~$F_n = \One_{[\frac{1}{n},\infty)}$.
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and $F_n$ the distribution function, i.e.~$F_n = \One_{[\frac{1}{n},\infty)}$.
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@ -67,8 +67,8 @@ In this lecture we recall the most important point from the lecture.
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inversion formula (\autoref{inversionformula}), ...
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inversion formula (\autoref{inversionformula}), ...
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\item Levy's continuity theorem (\autoref{levycontinuity}),
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\item Levy's continuity theorem (\autoref{levycontinuity}),
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(\autoref{genlevycontinuity})
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(\autoref{genlevycontinuity})
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\item Bockner's theorem for positive definite function % TODO REF
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\item Bochner's theorem for positive definite function (\autoref{thm:bochner})
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\item Bockner's theorem for the mass at a point (\autoref{bochnersformula})
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\item Bochner's theorem for the mass at a point (\autoref{bochnersformula})
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\item Related notions
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\item Related notions
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\todo{TODO}
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\todo{TODO}
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\begin{itemize}
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\begin{itemize}
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