boundedness
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Important stuff not done in the lecture.
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\section{Additional Material}
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Moments:
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% Important stuff not done in the lecture.
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$\bE[X^k]$
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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{lemma}
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Let $X, Y : \Omega \to [a,b]$
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If $\bE[X^k] = \bE[Y^k]$,
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for every $k \in \N_0$
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then $X = Y$.
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\end{lemma}
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\begin{proof}
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We have $\bE[p(X)] = \bE[p(Y)]$ for
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every polynomial $p$.
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Approximate $e^{\i t X}$
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with polynomials and use Fourier transforms.
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\end{proof}
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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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Laplace transforms
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\subsection{Laplace Transforms}
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\todo{Write something about Laplace Transforms}
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@ -22,7 +22,11 @@ First, let us recall some basic definitions:
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\end{itemize}
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\end{itemize}
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\end{definition}
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\begin{definition}+
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Let $X$ be a random variable and $k \in \N$.
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Then the $k$-th \vocab{moment} of $X$ is defined as
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$\bE[X^k]$.
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\end{definition}
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\begin{definition}
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A \vocab{random variable} $X : (\Omega, \cF) \to (\R, \cB(\R))$
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@ -63,6 +67,7 @@ The converse to this fact is also true:
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See theorem 2.4.3 in Stochastik.
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\end{proof}
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\begin{example}[Some important probability distribution functions]\hfill
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\begin{enumerate}[(1)]
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\item \vocab{Uniform distribution} on $[0,1]$:
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@ -161,7 +161,7 @@ More formally:
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The idea of `` $\implies$ '' will lead to coupling. % TODO ?
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\end{remark}
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A proof of \autoref{thm5} can be found in the notes.\notes
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\begin{example}[Application of \autoref{thm5}]
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\begin{example}[Application of \autoref{thm4}]
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The series $\sum_{n} \frac{1}{n^{\frac{1}{2} + \epsilon}}$
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does not converge for $\epsilon < \frac{1}{2}$.
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However
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@ -183,7 +183,7 @@ However, some subsets can be easily described, e.g.
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Since $\phi$ is Lipschitz,
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$ X_n \xrightarrow{\bP} X \implies \phi(X_n) \xrightarrow{\bP} \phi(X)$.
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By the bounded convergence theorem % TODO
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By the bounded convergence theorem, \autoref{thm:boundedconvergence},
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$|\phi(X_n)| \le k \implies \int | \phi(X_n) - \phi(X)| \dif \bP \to 0$.
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\subsection{Stopping Times}
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\begin{definition}[Stopping time]
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\label{def:stopping-time}
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A random variable $T: \Omega \to \N_0 \cup \{\infty\}$ on a filtered probability space $(\Omega, \cF, \{\cF_n\}_n, \bP)$ is called a \vocab{stopping time},
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if
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\[
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@ -151,9 +151,10 @@ In this lecture we recall the most important point from the lecture.
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\item This is an important proof.
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\end{itemize}
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\item Uniform integrability % TODO
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\item What are stopping times?
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\item What are stopping times? \autoref{def:stopping-time}
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\item (Non-)examples of stopping times
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\item \textbf{Optional stopping theorem} - be really comfortable with this.
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\autoref{optionalstopping}
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\end{itemize}
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@ -17,7 +17,7 @@
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\tableofcontents
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\cleardoublepage
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%\mainmatter
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%\mainatter
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\input{inputs/intro.tex}
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\section{Appendix}
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\input{inputs/a_0_distributions.tex}
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\end{landscape}
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\pagebreak
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\input{inputs/a_2_additional_stuff.tex}
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\input{inputs/lecture_23.tex}
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\cleardoublepage
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