From 9cf7536921c39e296cacf01049db11f61453577b Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Tue, 18 Jul 2023 23:58:27 +0200 Subject: [PATCH] summary --- inputs/lecture_16.tex | 11 +++++++++++ inputs/lecture_18.tex | 1 + inputs/lecture_19.tex | 2 ++ inputs/lecture_23.tex | 28 ++++++++++++++++++---------- 4 files changed, 32 insertions(+), 10 deletions(-) diff --git a/inputs/lecture_16.tex b/inputs/lecture_16.tex index 62dce27..f705564 100644 --- a/inputs/lecture_16.tex +++ b/inputs/lecture_16.tex @@ -74,6 +74,16 @@ The Radon Nikodym theorem is the converse of that: w.r.t.~$\mu$. This is written as $\nu \ll \mu$. \end{definition} +\begin{definition}+ + Two measures $\mu$ and $\nu$ on a measure space $(\Omega, \cF)$ + are called \vocab{singular}, + denoted $\mu \bot \nu$, + iff there exists $A \in \cF$ such that + \[ + \mu(A) = \nu(A^c) = 0. + \] +\end{definition} + With \autoref{radonnikodym} we get a very short proof of the existence of conditional expectation: @@ -171,6 +181,7 @@ we have gathered up to time $n$. Typically $\cF_n = \sigma(X_1, \ldots, X_n)$ for a sequence of random variables. \begin{definition} + \label{def:martingale} Let $(\cF_n)$ be a filtration and $X_1,\ldots,X_n$ be random variables such that $X_i \in L^1(\bP)$. Then we say that $(X_n)_{n \ge 1}$ is an $(\cF_n)_n$-\vocab{martingale} diff --git a/inputs/lecture_18.tex b/inputs/lecture_18.tex index 11fafab..4df42f5 100644 --- a/inputs/lecture_18.tex +++ b/inputs/lecture_18.tex @@ -20,6 +20,7 @@ Hence the same holds for submartingales, i.e. \end{question} \begin{example}[A martingale not converging in $L^1$ ] + \label{ex:martingale-not-converging-in-l1} Fix $u > 1$ and let $p = \frac{1}{1+u}$. Let $ (Z_n)_{n \ge 1}$ be i.i.d.~$\pm 1$ with $\bP[Z_n = 1] = p$. diff --git a/inputs/lecture_19.tex b/inputs/lecture_19.tex index d369715..6df66a0 100644 --- a/inputs/lecture_19.tex +++ b/inputs/lecture_19.tex @@ -23,6 +23,7 @@ \end{goal} \begin{definition} + \label{def:ui} A sequence of random variables $(X_n)_n$ is called \vocab{uniformly integrable} (UI), if \[\forall \epsilon > 0 .~\exists K > 0 .~ \forall n.~ @@ -245,4 +246,5 @@ Let $(\Omega, \cF, \bP)$ as always and let $(\cF_n)_n$ always be a filtration. Let $(X_n)_n$ be a martingale bounded in $L^p$. Then there exists a random variable $X \in L^p$, such that $X_n = \bE[X | \cF_n]$ for all $n$. + In particular, $X_n \xrightarrow{L^p} X$. \end{theorem} diff --git a/inputs/lecture_23.tex b/inputs/lecture_23.tex index dcb5536..60b08d7 100644 --- a/inputs/lecture_23.tex +++ b/inputs/lecture_23.tex @@ -117,7 +117,7 @@ In this lecture we recall the most important point from the lecture. \begin{itemize} \item iid (\autoref{clt}), \item Lindeberg (\autoref{lindebergclt}), - \item Luyapanov (\autoref{lyapunovclt}) + \item Lyapanov (\autoref{lyapunovclt}) \end{itemize} \item How to apply this? Exercises! \end{itemize} @@ -125,10 +125,12 @@ In this lecture we recall the most important point from the lecture. \subsubsection{Conditional expectation} \begin{itemize} \item Definition and existence of conditional expectation for $X \in L^1(\Omega, \cF, \bP)$ + (\autoref{conditionalexpectation}) \item If $H = L^2(\Omega, \cF, \bP)$, then $\bE[ \cdot | \cG]$ is the (unique) projection on the closed subspace $L^2(\Omega, \cG, \bP)$. Why is this a closed subspace? Why is the projection orthogonal? - \item Radon-Nikodym Theorem (Proof not relevant for the exam) + \item Radon-Nikodym Theorem \ref{radonnikodym} + (Proof not relevant for the exam) \item (Non-)examples of mutually absolutely continuous measures Singularity in this context? % TODO \end{itemize} @@ -136,25 +138,31 @@ In this lecture we recall the most important point from the lecture. \subsubsection{Martingales} \begin{itemize} - \item Definition of Martingales - \item Doob's convergence theorem, Upcrossing inequality + \item Definition of Martingales (\autoref{def:martingale}) + \item Doob's convergence theorem (\autoref{doobmartingaleconvergence}), + Upcrossing inequality (\autoref{lec17l1}, \autoref{lec17l2}, \autoref{lec17l3}) (downcrossings for submartingales) \item Examples of Martingales converging a.s.~but not in $L^1$ - \item Bounded in $L^2$ $\implies$ convergence in $L^2$. + (\autoref{ex:martingale-not-converging-in-l1}) + \item Bounded in $L^2$ $\implies$ convergence in $L^2$ + (\autoref{martingaleconvergencel2}). \item Martingale increments are orthogonal in $L^2$! + (\autoref{martingaleincrementsorthogonal}) \item Doob's (sub-)martingale inequalities + (\autoref{dooblp}), \item $\bP[\sup_{k \le n} M_k \ge x]$ $\leadsto$ Look at martingale inequalities! Estimates might come from Doob's inequalities if $(M_k)_k$ is a (sub-)martingale. - \item Doob's $L^p$ convergence theorem. + \item Doob's $L^p$ convergence theorem + (\autoref{ceismartingale}, \autoref{martingaleisce}). \begin{itemize} \item Why is $p > 1$ important? \textbf{Role of Banach-Alaoglu} \item This is an important proof. \end{itemize} - \item Uniform integrability % TODO - \item What are stopping times? \autoref{def:stopping-time} + \item Uniform integrability (\autoref{def:ui}) + \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} + \item \textbf{Optional stopping theorem} (\autoref{optionalstopping}) + - be really comfortable with this. \end{itemize}