some facts
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@ -83,6 +83,35 @@ This will be the weakest notion of convergence, hence it is called
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This notion of convergence will be defined in terms of
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This notion of convergence will be defined in terms of
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characteristic functions of Fourier transforms.
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characteristic functions of Fourier transforms.
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\subsection{Convolutions${}^\dagger$}
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\begin{definition}+[Convolution]
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Let $\mu$ and $\nu$ be probability measures on $\R^d$
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with Lebesgue densities $f_\mu$ and $f_\nu$.
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Then the \vocab{convolution} of $\mu$ and $\nu$,
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$\mu \ast \nu$,
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is the probability measure on $\R^d$
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with Lebesgue density
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\[
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f_{\mu \ast \nu}(x) \coloneqq
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\int_{\R^d} f_\mu(x - t) f\nu(t) \lambda^d(\dif t)
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\]
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\end{definition}
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\begin{fact}+[Exercise 6.1]
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If $X_1,X_2,\ldots$ are independent with
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distributions $X_1 \sim \mu_1$,
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$X_2 \sim \mu_2, \ldots$,
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then $X_1 + \ldots + X_n$
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has distribution
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\[
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\mu_1 \ast \mu_2 \ast \ldots \ast \mu_n.
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\]
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\end{fact}
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\todo{TODO}
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\subsection{Characteristic Functions and Fourier Transform}
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\subsection{Characteristic Functions and Fourier Transform}
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\begin{definition}
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\begin{definition}
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@ -106,7 +135,30 @@ We have
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\item We have $\phi(0) = 1$.
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\item We have $\phi(0) = 1$.
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\item $|\phi(t)| \le \int_{\R} |e^{\i t x} | \bP(dx) = 1$.
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\item $|\phi(t)| \le \int_{\R} |e^{\i t x} | \bP(dx) = 1$.
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\end{itemize}
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\end{itemize}
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\todo{Properties of characteristic functions}
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\begin{fact}+
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Let $X$, $Y$ be independent random variables
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and $a,b \in \R$.
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Then
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\begin{itemize}
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\item $\phi_{a X + b}(t) = e^{\i t b} \phi_X(\frac{t}{a})$,
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\item $\phi_{X + Y}(t) = \phi_X(t) + \phi_Y(t)$.
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\end{itemize}
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\end{fact}
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\begin{proof}
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We have
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\begin{IEEEeqnarray*}{rCl}
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\phi_{a X + b}(t) &=& \bE[e^{\i t (aX + b)}}]\\
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&=& e^{\i t b} \bE[e^{\i t a X}]\\
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&=& e^{\i t b} \phi_X(\frac{t}{a}).
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\end{IEEEeqnarray*}
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Furthermore
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\begin{IEEEeqnarray*}{rCl}
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\phi_{X + Y}(t) &=& \bE[e^{\i t (X + Y)}]\\
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&=& \bE[e^{\i t X}] \bE[e^{\i t Y}]\\
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&=& \phi_X(t) \phi_Y(t).
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\end{IEEEeqnarray*}
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\end{proof}
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\begin{remark}
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\begin{remark}
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Suppose $(\Omega, \cF, \bP)$ is an arbitrary probability space and
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Suppose $(\Omega, \cF, \bP)$ is an arbitrary probability space and
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@ -1,7 +1,5 @@
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\lecture{10}{2023-05-09}{}
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\lecture{10}{2023-05-09}{}
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% RECAP
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First, we will prove some of the most important facts about Fourier transforms.
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First, we will prove some of the most important facts about Fourier transforms.
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We consider $(\R, \cB(\R))$.
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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.
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\item Non-examples: $(\delta_n)_n$
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\item Non-examples: $(\delta_n)_n$
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\item How does one prove weak convergence? How does one write this down in a clear way?
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\item How does one prove weak convergence? How does one write this down in a clear way?
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% TODO
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\begin{itemize}
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\item \autoref{lec10_thm1},
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\item Levy's continuity theorem
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\ref{levycontinuity},
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\item Generalization of Levy's continuity theorem
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\ref{genlevycontinuity}
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\end{itemize}
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\end{itemize}
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\end{itemize}
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\paragraph{Convolution}
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\paragraph{Convolution}
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@ -104,7 +110,6 @@ In this lecture we recall the most important point from the lecture.
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\end{itemize}
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\end{itemize}
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\subsubsubsection{CLT}
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\subsubsubsection{CLT}
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\begin{itemize}
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\begin{itemize}
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\item Statement of the CLT
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\item Statement of the CLT
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