josia-notes/s23-probability-theory
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fixed typos

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Josia Pietsch 2023-07-28 03:56:43 +02:00
parent 296c7b2f55
commit 9d1d53877a
Signed by: josia
GPG key ID: E70B571D66986A2D
3 changed files with 5 additions and 5 deletions
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6
inputs/lecture_12.tex
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@ -29,14 +29,14 @@ First, we need to prove some properties of characteristic functions.
\begin{refproof}{charfprops} \begin{refproof}{charfprops}
\begin{enumerate}[(i)] \begin{enumerate}[(i)]
\item $\phi_X(0) = \bE[e^{\i 0 X}] = \bE[1] = 1$. \item $\phi_X(0) = \bE[e^{\i 0 X}] = \bE[1] = 1$.
For $t \in \R$, we have $|\phi_X(t)| = |\bE[e^{\i t X}]| \overset{\text{Jensen}}{\le} \bE[|e^{\i t X}|] = 1$. For $t \in \R$, we have $|\phi_X(t)| = |\bE[e^{\i t X}]| \overset{\yaref{jensen}}{\le} \bE[|e^{\i t X}|] = 1$.
\item Let $t, h \in \R$. \item Let $t, h \in \R$.
Then Then
\begin{IEEEeqnarray*}{rCl} \begin{IEEEeqnarray*}{rCl}
|\phi_X(t+h) - \phi_X(t)| &=& |\bE[e^{\i (t+h) X} - e^{\i t X}]|\\ |\phi_X(t+h) - \phi_X(t)| &=& |\bE[e^{\i (t+h) X} - e^{\i t X}]|\\
&=& |\bE[e^{\i t X} (e^{\i h X} - 1)]|\\ &=& |\bE[e^{\i t X} (e^{\i h X} - 1)]|\\
&\overset{\text{Jensen}}{\le}& &\overset{\yaref{jensen}}{\le}&
\bE[|e^{\i t X}| \cdot |e^{\i h X} -1|]\\ \bE[|e^{\i t X}| \cdot |e^{\i h X} -1|]\\
&=& \bE[|e^{\i h X} - 1|] &=& \bE[|e^{\i h X} - 1|]
\text{\reflectbox{$\coloneqq$}} g(h)\\ \text{\reflectbox{$\coloneqq$}} g(h)\\
@ -73,7 +73,7 @@ First, we need to prove some properties of characteristic functions.
\begin{IEEEeqnarray*}{rCl} \begin{IEEEeqnarray*}{rCl}
|e^{\i y} - 1| &=& |\int_0^y \cos(s) \dif s + \i \int_0^y \sin(s) \dif s|\\ |e^{\i y} - 1| &=& |\int_0^y \cos(s) \dif s + \i \int_0^y \sin(s) \dif s|\\
&=& |\int_0^y e^{\i s} \dif s|\\ &=& |\int_0^y e^{\i s} \dif s|\\
&\overset{\text{Jensen}}{\le}& \int_0^y |e^{\i s}| ds = y. &\overset{\yaref{jensen}}{\le}& \int_0^y |e^{\i s}| ds = y.
\end{IEEEeqnarray*} \end{IEEEeqnarray*}
For $y < 0$, we have $|e^{\i y} - 1| = |e^{-\i y} - 1|$ For $y < 0$, we have $|e^{\i y} - 1| = |e^{-\i y} - 1|$
and we can apply the above to $-y$. and we can apply the above to $-y$.

2
inputs/lecture_19.tex
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@ -61,7 +61,7 @@ However, some subsets can be easily described, e.g.
Since $\sup_n \bE[|X_n|^{1+\delta}] < \infty$, Since $\sup_n \bE[|X_n|^{1+\delta}] < \infty$,
we have that $\sup_n \bE[|X_n|] < \infty$ by \yaref{jensen}. we have that $\sup_n \bE[|X_n|] < \infty$ by \yaref{jensen}.
Hence for $K$ large enough relevant term is less than $\epsilon$. Hence for $K$ large enough the relevant term is less than $\epsilon$.
\end{proof} \end{proof}
\begin{fact}\label{lec19f2} \begin{fact}\label{lec19f2}

2
inputs/lecture_20.tex
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@ -60,7 +60,7 @@
\begin{IEEEeqnarray*}{rCl} \begin{IEEEeqnarray*}{rCl}
\|X_n - X_n'\|_{L^p}^p \|X_n - X_n'\|_{L^p}^p
&=& \bE[\bE[X - X' | \cF_n]^{p}]\\ &=& \bE[\bE[X - X' | \cF_n]^{p}]\\
&\overset{\text{Jensen}}{\le}& \bE[\bE[(X - X')^p | \cF_n]]\\ &\overset{\yaref{cjensen}}{\le}& \bE[\bE[(X - X')^p | \cF_n]]\\
&=& \|X - X'\|_{L^p}^p\\ &=& \|X - X'\|_{L^p}^p\\
&<& \epsilon. &<& \epsilon.
\end{IEEEeqnarray*} \end{IEEEeqnarray*}