65 lines
2.5 KiB
TeX
65 lines
2.5 KiB
TeX
% \subsection{Additional Material}
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% Important stuff not done in the lecture.
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\subsection{Notions of boundedness}
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The following is just a short overview of all the notions of
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boundedness we used in the lecture.
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\begin{definition}+[Boundedness]
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Let $\cX$ be a set of random variables.
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We say that $\cX$ is
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\begin{itemize}
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\item \vocab{uniformly bounded} iff
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\[\sup_{X \in \cX} \sup_{\omega \in \Omega} |X(\omega)| < \infty,\]
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\item \vocab{dominated by $f \in L^p$} for $p \ge 1$ iff
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\[
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\forall X \in \cX .~ |X| \le f,
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\]
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\item \vocab{bounded in $L^p$} for $p \ge 1$ iff
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\[
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\sup_{X \in \cX} \|X\|_{L^p} < \infty,
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\]
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\item \vocab{uniformly integrable} iff
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\[
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\forall \epsilon > 0 .~\exists K .~ \forall X \in \cX.~
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\bE[|X| \One_{|X| > K}] < \epsilon.
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\]
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\end{itemize}
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\end{definition}
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\begin{fact}+
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Let $\cX$ be a set of random variables.
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Let $1 < p \le q < \infty$
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Then the following implications hold:
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\begin{figure}[H]
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\centering
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\begin{tikzpicture}
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\node at (0,2.5) (ub) {$\cX$ is uniformly bounded};
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\node at (-2.5,1.5) (dq) {$\cX$ is dominated by $f \in L^q$};
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\node at (-2.5,0.5) (dp) {$\cX$ is dominated by $f \in L^p$};
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\node at (2.5,1.0) (bq) {$\cX$ is bounded in $L^q$};
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\node at (2.5,0) (bp) {$\cX$ is bounded in $L^p$};
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\node at (-2.5,-0.5) (d1) {$\cX$ is dominated by $f \in L^1$};
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\node at (0,-1.5) (ui) {$\cX$ is uniformly integrable};
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\node at (2.5,-2.5) (b1) {$\cX$ is bounded in $L^1$};
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\draw[double equal sign distance, -implies] (ub) -- (dq);
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% \draw[double equal sign distance, -implies] (ub) -- (bq);
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\draw[double equal sign distance, -implies] (bq) -- (bp);
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\draw[double equal sign distance, -implies] (dq) -- (dp);
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\draw[double equal sign distance, -implies] (dq) -- (bq);
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\draw[double equal sign distance, -implies] (dp) -- (bp);
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\draw[double equal sign distance, -implies] (bp) -- (ui);
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\draw[double equal sign distance, -implies] (dp) -- (d1);
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\draw[double equal sign distance, -implies] (d1) -- (ui);
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\draw[double equal sign distance, -implies] (ui) -- (b1);
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\end{tikzpicture}
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\end{figure}
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\end{fact}
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\subsection{Laplace Transforms}
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\todo{Write something about Laplace Transforms}
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