From fae60388abf69ea205166035a7fa7b0a1bf50180 Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Tue, 18 Jul 2023 22:33:10 +0200 Subject: [PATCH] boundedness --- inputs/a_2_additional_stuff.tex | 77 ++++++++++++++++++++++++++------- inputs/lecture_01.tex | 7 ++- inputs/lecture_07.tex | 2 +- inputs/lecture_19.tex | 2 +- inputs/lecture_20.tex | 1 + inputs/lecture_23.tex | 3 +- probability_theory.tex | 4 +- 7 files changed, 75 insertions(+), 21 deletions(-) diff --git a/inputs/a_2_additional_stuff.tex b/inputs/a_2_additional_stuff.tex index 95a734d..8d7f910 100644 --- a/inputs/a_2_additional_stuff.tex +++ b/inputs/a_2_additional_stuff.tex @@ -1,21 +1,66 @@ -Important stuff not done in the lecture. +\section{Additional Material} -Moments: +% Important stuff not done in the lecture. -$\bE[X^k]$ +\subsection{Notions of boundedness} +The following is just a short overview of all the notions of +boundedness we used in the lecture. -\begin{lemma} - Let $X, Y : \Omega \to [a,b]$ - If $\bE[X^k] = \bE[Y^k]$, - for every $k \in \N_0$ - then $X = Y$. -\end{lemma} -\begin{proof} - We have $\bE[p(X)] = \bE[p(Y)]$ for - every polynomial $p$. - Approximate $e^{\i t X}$ - with polynomials and use Fourier transforms. -\end{proof} +\begin{definition}+[Boundedness] + Let $\cX$ be a set of random variables. + We say that $\cX$ is + \begin{itemize} + \item \vocab{uniformly bounded} iff + \[\sup_{X \in \cX} \sup_{\omega \in \Omega} |X(\omega)| < \infty,\] + \item \vocab{dominated by $f \in L^p$} for $p \ge 1$ iff + \[ + \forall X \in \cX .~ |X| \le f, + \] + \item \vocab{bounded in $L^p$} for $p \ge 1$ iff + \[ + \sup_{X \in \cX} \|X\|_{L^p} < \infty, + \] + \item \vocab{uniformly integrable} iff + \[ + \forall \epsilon > 0 .~\exists K .~ \forall X \in \cX.~ + \bE[|X| \One_{|X| > K}] < \epsilon. + \] + \end{itemize} + +\end{definition} + +\begin{fact}+ + Let $\cX$ be a set of random variables. + Let $1 < p \le q < \infty$ + Then the following implications hold: + \begin{figure}[H] + \centering + \begin{tikzpicture} + \node at (0,2.5) (ub) {$\cX$ is uniformly bounded}; + \node at (-2.5,1.5) (dq) {$\cX$ is dominated by $f \in L^q$}; + \node at (-2.5,0.5) (dp) {$\cX$ is dominated by $f \in L^p$}; + \node at (2.5,1.0) (bq) {$\cX$ is bounded in $L^q$}; + \node at (2.5,0) (bp) {$\cX$ is bounded in $L^p$}; + \node at (-2.5,-0.5) (d1) {$\cX$ is dominated by $f \in L^1$}; + \node at (0,-1.5) (ui) {$\cX$ is uniformly integrable}; + \node at (2.5,-2.5) (b1) {$\cX$ is bounded in $L^1$}; + \draw[double equal sign distance, -implies] (ub) -- (dq); + % \draw[double equal sign distance, -implies] (ub) -- (bq); + \draw[double equal sign distance, -implies] (bq) -- (bp); + \draw[double equal sign distance, -implies] (dq) -- (dp); + \draw[double equal sign distance, -implies] (dq) -- (bq); + \draw[double equal sign distance, -implies] (dp) -- (bp); + \draw[double equal sign distance, -implies] (bp) -- (ui); + \draw[double equal sign distance, -implies] (dp) -- (d1); + \draw[double equal sign distance, -implies] (d1) -- (ui); + \draw[double equal sign distance, -implies] (ui) -- (b1); + \end{tikzpicture} + \end{figure} +\end{fact} -Laplace transforms + + + +\subsection{Laplace Transforms} +\todo{Write something about Laplace Transforms} diff --git a/inputs/lecture_01.tex b/inputs/lecture_01.tex index 5fa8519..4d2b019 100644 --- a/inputs/lecture_01.tex +++ b/inputs/lecture_01.tex @@ -22,7 +22,11 @@ First, let us recall some basic definitions: \end{itemize} \end{itemize} \end{definition} - +\begin{definition}+ + Let $X$ be a random variable and $k \in \N$. + Then the $k$-th \vocab{moment} of $X$ is defined as + $\bE[X^k]$. +\end{definition} \begin{definition} A \vocab{random variable} $X : (\Omega, \cF) \to (\R, \cB(\R))$ @@ -63,6 +67,7 @@ The converse to this fact is also true: See theorem 2.4.3 in Stochastik. \end{proof} + \begin{example}[Some important probability distribution functions]\hfill \begin{enumerate}[(1)] \item \vocab{Uniform distribution} on $[0,1]$: diff --git a/inputs/lecture_07.tex b/inputs/lecture_07.tex index 34a0d0b..c63f99a 100644 --- a/inputs/lecture_07.tex +++ b/inputs/lecture_07.tex @@ -161,7 +161,7 @@ More formally: The idea of `` $\implies$ '' will lead to coupling. % TODO ? \end{remark} A proof of \autoref{thm5} can be found in the notes.\notes -\begin{example}[Application of \autoref{thm5}] +\begin{example}[Application of \autoref{thm4}] The series $\sum_{n} \frac{1}{n^{\frac{1}{2} + \epsilon}}$ does not converge for $\epsilon < \frac{1}{2}$. However diff --git a/inputs/lecture_19.tex b/inputs/lecture_19.tex index 1dd0f4e..d369715 100644 --- a/inputs/lecture_19.tex +++ b/inputs/lecture_19.tex @@ -183,7 +183,7 @@ However, some subsets can be easily described, e.g. Since $\phi$ is Lipschitz, $ X_n \xrightarrow{\bP} X \implies \phi(X_n) \xrightarrow{\bP} \phi(X)$. - By the bounded convergence theorem % TODO + By the bounded convergence theorem, \autoref{thm:boundedconvergence}, $|\phi(X_n)| \le k \implies \int | \phi(X_n) - \phi(X)| \dif \bP \to 0$. diff --git a/inputs/lecture_20.tex b/inputs/lecture_20.tex index 00827ad..0afefa6 100644 --- a/inputs/lecture_20.tex +++ b/inputs/lecture_20.tex @@ -122,6 +122,7 @@ we need the following theorem, which we won't prove here: \subsection{Stopping Times} \begin{definition}[Stopping time] + \label{def:stopping-time} 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}, if \[ diff --git a/inputs/lecture_23.tex b/inputs/lecture_23.tex index 50cf647..dcb5536 100644 --- a/inputs/lecture_23.tex +++ b/inputs/lecture_23.tex @@ -151,9 +151,10 @@ In this lecture we recall the most important point from the lecture. \item This is an important proof. \end{itemize} \item Uniform integrability % TODO - \item What are stopping times? + \item What are stopping times? \autoref{def:stopping-time} \item (Non-)examples of stopping times \item \textbf{Optional stopping theorem} - be really comfortable with this. + \autoref{optionalstopping} \end{itemize} diff --git a/probability_theory.tex b/probability_theory.tex index c398eb6..011f427 100644 --- a/probability_theory.tex +++ b/probability_theory.tex @@ -17,7 +17,7 @@ \tableofcontents \cleardoublepage -%\mainmatter +%\mainatter \input{inputs/intro.tex} @@ -55,6 +55,8 @@ \section{Appendix} \input{inputs/a_0_distributions.tex} \end{landscape} +\pagebreak +\input{inputs/a_2_additional_stuff.tex} \input{inputs/lecture_23.tex} \cleardoublepage