josia-notes/s23-probability-theory
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fixed small errors

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Josia Pietsch 2023-05-11 23:55:51 +02:00
parent 41cb2f2094
commit db99990b63
Signed by untrusted user who does not match committer: josia
GPG Key ID: E70B571D66986A2D
2 changed files with 8 additions and 4 deletions
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8
inputs/lecture_10.tex
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@ -27,10 +27,10 @@ where $\mu = \bP X^{-1}$.
We have
\begin{IEEEeqnarray*}{rCl}
&&\lim_{T \to \infty} \frac{1}{2 \pi} \int_{-T}^T \int_{\R} \frac{e^{-\i t b}- e^{-\i t a}}{-\i t} e^{\i t x} d \bP(x)\\
&\overset{\text{Fubini for $L^1$}}{=}& \lim_{T \to \infty} \frac{1}{2 \pi} \int_{\R} \int_{-T}^T \frac{e^{-\i t b}- e^{-\i t a}}{-\i t} e^{\i t x} d \bP(x)\\
&=& \lim_{T \to \infty} \frac{1}{2 \pi} \int_{\R} \int_{-T}^T \frac{e^{\i t (b-x)}- e^{\i t (x-a)}}{-\i t} d \bP(x)\\
&=& \lim_{T \to \infty} \frac{1}{2 \pi} \int_{\R} \underbrace{\int_{-T}^T \left[ \frac{\cos(t (x-b)) - \cos(t(x-a))}{-\i t}\right] d \bP(x)}_{=0 \text{, as the function is odd}}
&&\lim_{T \to \infty} \frac{1}{2 \pi} \int_{-T}^T \int_{\R} \frac{e^{-\i t b}- e^{-\i t a}}{-\i t} e^{\i t x} dt d \bP(x)\\
&\overset{\text{Fubini for $L^1$}}{=}& \lim_{T \to \infty} \frac{1}{2 \pi} \int_{\R} \int_{-T}^T \frac{e^{-\i t b}- e^{-\i t a}}{-\i t} e^{\i t x} dt d \bP(x)\\
&=& \lim_{T \to \infty} \frac{1}{2 \pi} \int_{\R} \int_{-T}^T \frac{e^{\i t (b-x)}- e^{\i t (x-a)}}{-\i t} dt d \bP(x)\\
&=& \lim_{T \to \infty} \frac{1}{2 \pi} \int_{\R} \underbrace{\int_{-T}^T \left[ \frac{\cos(t (x-b)) - \cos(t(x-a))}{-\i t}\right] dt d \bP(x)}_{=0 \text{, as the function is odd}}
\\&&
+ \lim_{T \to \infty} \frac{1}{2\pi} \int_{\R}\int_{-T}^T \frac{\sin(t ( x - b)) - \sin(t(x-a))}{-t} dt d\bP(x)\\
&=& \lim_{T \to \infty} \frac{1}{\pi} \int_\R \int_{0}^T \frac{\sin(t(x-a)) - \sin(t(x-b))}{t} dt d\bP(x)\\

4
wtheo.sty
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@ -93,3 +93,7 @@
\NewFancyTheorem[thmtools = { style = thmredmargin} , group = { big } ]{warning}
\DeclareSimpleMathOperator{Var}
\DeclareSimpleMathOperator{Bin}
\DeclareSimpleMathOperator{Ber}
\DeclareSimpleMathOperator{Exp}