From db99990b6370e6b1a8e11a84bf01a80b2c045b18 Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Thu, 11 May 2023 23:55:51 +0200 Subject: [PATCH] fixed small errors --- inputs/lecture_10.tex | 8 ++++---- wtheo.sty | 4 ++++ 2 files changed, 8 insertions(+), 4 deletions(-) diff --git a/inputs/lecture_10.tex b/inputs/lecture_10.tex index 2a6f4ea..a37708b 100644 --- a/inputs/lecture_10.tex +++ b/inputs/lecture_10.tex @@ -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)\\ diff --git a/wtheo.sty b/wtheo.sty index 14a67cf..728d857 100644 --- a/wtheo.sty +++ b/wtheo.sty @@ -93,3 +93,7 @@ \NewFancyTheorem[thmtools = { style = thmredmargin} , group = { big } ]{warning} \DeclareSimpleMathOperator{Var} + +\DeclareSimpleMathOperator{Bin} +\DeclareSimpleMathOperator{Ber} +\DeclareSimpleMathOperator{Exp}