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Josia Pietsch 2023-07-10 12:20:52 +02:00
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26
inputs/lecture_23.tex
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@ -1,8 +1,8 @@
\lecture{23}{2023-07-06}{} \lecture{23}{2023-07-06}{Recap}
\section{Recap} \subsection{Recap}
In this lecture we will recall the most important point from the lecture. In this lecture we recall the most important point from the lecture.
\subsection{Construction of iid random variables.} \subsubsection{Construction of iid random variables.}
\begin{itemize} \begin{itemize}
\item Definition of a consistent family (\autoref{def:consistentfamily}) \item Definition of a consistent family (\autoref{def:consistentfamily})
@ -20,20 +20,18 @@ In this lecture we will recall the most important point from the lecture.
(\autoref{thm:kolmogorovconsistency}) (\autoref{thm:kolmogorovconsistency})
\end{itemize} \end{itemize}
\subsection{Limit theorems} \subsubsection{Limit theorems}
\begin{itemize} \begin{itemize}
\item Work with iid.~random variables. \item Work with iid.~random variables.
\item Notions of convergence (\autoref{def:convergence}) \item Notions of convergence (\autoref{def:convergence})
\item Implications between different notions of convergence (very important) and counter examples. \item Implications between different notions of convergence (very important) and counter examples.
(\autoref{thm:convergenceimplications}) (\autoref{thm:convergenceimplications})
\item \begin{itemize} \item \item Laws of large numbers: (\autoref{lln})
\item Laws of large numbers: (\autoref{lln})
\begin{itemize} \begin{itemize}
\item WLLN: convergence in probability \item WLLN: convergence in probability
\item SLLN: weak convergence \item SLLN: weak convergence
\end{itemize} \end{itemize}
\end{itemize}
\item \autoref{thm2} (building block for SLLN): \item \autoref{thm2} (building block for SLLN):
Let $(X_n)$ be independent with mean $0$ and $\sum \sigma_n^2 < \infty$, Let $(X_n)$ be independent with mean $0$ and $\sum \sigma_n^2 < \infty$,
then $ \sum X_n $ converges a.s. then $ \sum X_n $ converges a.s.
@ -58,7 +56,7 @@ In this lecture we will recall the most important point from the lecture.
\end{itemize} \end{itemize}
\end{itemize} \end{itemize}
\subsubsection{Fourier transform / characteristic functions / weak convegence} \subsubsubsection{Fourier transform / characteristic functions / weak convegence}
\begin{itemize} \begin{itemize}
\item Definition of Fourier transform \item Definition of Fourier transform
@ -101,13 +99,13 @@ In this lecture we will recall the most important point from the lecture.
\paragraph{Convolution} \paragraph{Convolution}
\begin{itemize} \begin{itemize}
\item Definition of convolution. \item Definition of convolution.
\todo{Copy from exercise sheet and write a section about this} \todo{Copy from exercise sheet and write a subsection about this}
\item $X_i \sim \mu_i \text{ iid. }\implies X_1 + \ldots + X_n \sim \mu_1 \ast \ldots \ast \mu_n$. \item $X_i \sim \mu_i \text{ iid. }\implies X_1 + \ldots + X_n \sim \mu_1 \ast \ldots \ast \mu_n$.
\end{itemize} \end{itemize}
\subsubsection{CLT} \subsubsubsection{CLT}
\begin{itemize} \begin{itemize}
\item Statement of the CLT \item Statement of the CLT
\item Several versions: \item Several versions:
@ -119,7 +117,7 @@ In this lecture we will recall the most important point from the lecture.
\item How to apply this? Exercises! \item How to apply this? Exercises!
\end{itemize} \end{itemize}
\subsection{Conditional expectation} \subsubsection{Conditional expectation}
\begin{itemize} \begin{itemize}
\item Definition and existence of conditional expectation for $X \in L^1(\Omega, \cF, \bP)$ \item Definition and existence of conditional expectation for $X \in L^1(\Omega, \cF, \bP)$
\item If $H = L^2(\Omega, \cF, \bP)$, then $\bE[ \cdot | \cG]$ \item If $H = L^2(\Omega, \cF, \bP)$, then $\bE[ \cdot | \cG]$
@ -130,7 +128,7 @@ In this lecture we will recall the most important point from the lecture.
Singularity in this context? % TODO Singularity in this context? % TODO
\end{itemize} \end{itemize}
\subsection{Martingales} \subsubsection{Martingales}
\begin{itemize} \begin{itemize}
\item Definition of Martingales \item Definition of Martingales
@ -154,7 +152,7 @@ In this lecture we will recall the most important point from the lecture.
\end{itemize} \end{itemize}
\subsection{Markov Chains} \subsubsection{Markov Chains}
\begin{itemize} \begin{itemize}
\item What are Markov chains? \item What are Markov chains?

6
inputs/prerequisites.tex
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@ -73,7 +73,9 @@ from the lecture on stochastic.
\begin{subproof} \begin{subproof}
We have $\bE[|X_n - X|] \to 0$. We have $\bE[|X_n - X|] \to 0$.
Suppose there exists an $\epsilon > 0$ such that Suppose there exists an $\epsilon > 0$ such that
$\lim_{n \to \infty} \bP[|X_n - X| > \epsilon] = c > 0$. $\limsup\limits_{n \to \infty} \bP[|X_n - X| > \epsilon] = c > 0$.
W.l.o.g.~$\lim_{n \to \infty} \bP[|X_n - X| > \epsilon] = c$,
otherwise choose an appropriate subsequence.
We have We have
\begin{IEEEeqnarray*}{rCl} \begin{IEEEeqnarray*}{rCl}
\bE[|X_n - X|] &=& \int_\Omega |X_n - X | d\bP\\ \bE[|X_n - X|] &=& \int_\Omega |X_n - X | d\bP\\
@ -90,7 +92,7 @@ from the lecture on stochastic.
$X_n \xrightarrow{\bP} X \notimplies X_n\xrightarrow{L^1} X$ $X_n \xrightarrow{\bP} X \notimplies X_n\xrightarrow{L^1} X$
\end{claim} \end{claim}
\begin{subproof} \begin{subproof}
Take $([0,1], \cB([0,1 ]), \lambda)([0,1], \cB([0,1 ]), \lambda)$ Take $([0,1], \cB([0,1 ]), \lambda)$
and define $X_n \coloneqq n \One_{[0, \frac{1}{n}]}$. and define $X_n \coloneqq n \One_{[0, \frac{1}{n}]}$.
We have $\bP[|X_n| > \epsilon] = \frac{1}{n}$ We have $\bP[|X_n| > \epsilon] = \frac{1}{n}$
for $n$ large enough. for $n$ large enough.