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\lecture{23}{2023-07-06}{}
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\section{Recap}
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In this lecture we will recall the most important point from the lecture.
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\lecture{23}{2023-07-06}{Recap}
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\subsection{Recap}
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In this lecture we recall the most important point from the lecture.
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\subsection{Construction of iid random variables.}
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\subsubsection{Construction of iid random variables.}
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\begin{itemize}
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\item Definition of a consistent family (\autoref{def:consistentfamily})
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(\autoref{thm:kolmogorovconsistency})
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\end{itemize}
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\subsection{Limit theorems}
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\subsubsection{Limit theorems}
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\begin{itemize}
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\item Work with iid.~random variables.
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\item Notions of convergence (\autoref{def:convergence})
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\item Implications between different notions of convergence (very important) and counter examples.
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(\autoref{thm:convergenceimplications})
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\item \begin{itemize}
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\item Laws of large numbers: (\autoref{lln})
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\item \item Laws of large numbers: (\autoref{lln})
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\begin{itemize}
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\item WLLN: convergence in probability
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\item SLLN: weak convergence
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\end{itemize}
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\end{itemize}
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\item \autoref{thm2} (building block for SLLN):
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Let $(X_n)$ be independent with mean $0$ and $\sum \sigma_n^2 < \infty$,
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then $ \sum X_n $ converges a.s.
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\end{itemize}
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\end{itemize}
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\subsubsection{Fourier transform / characteristic functions / weak convegence}
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\subsubsubsection{Fourier transform / characteristic functions / weak convegence}
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\begin{itemize}
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\item Definition of Fourier transform
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@ -101,13 +99,13 @@ In this lecture we will recall the most important point from the lecture.
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\paragraph{Convolution}
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\begin{itemize}
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\item Definition of convolution.
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\todo{Copy from exercise sheet and write a section about this}
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\todo{Copy from exercise sheet and write a subsection about this}
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\item $X_i \sim \mu_i \text{ iid. }\implies X_1 + \ldots + X_n \sim \mu_1 \ast \ldots \ast \mu_n$.
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\end{itemize}
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\subsubsection{CLT}
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\subsubsubsection{CLT}
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\begin{itemize}
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\item Statement of the CLT
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\item Several versions:
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@ -119,7 +117,7 @@ In this lecture we will recall the most important point from the lecture.
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\item How to apply this? Exercises!
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\end{itemize}
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\subsection{Conditional expectation}
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\subsubsection{Conditional expectation}
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\begin{itemize}
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\item Definition and existence of conditional expectation for $X \in L^1(\Omega, \cF, \bP)$
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\item If $H = L^2(\Omega, \cF, \bP)$, then $\bE[ \cdot | \cG]$
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Singularity in this context? % TODO
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\end{itemize}
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\subsection{Martingales}
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\subsubsection{Martingales}
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\begin{itemize}
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\item Definition of Martingales
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\end{itemize}
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\subsection{Markov Chains}
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\subsubsection{Markov Chains}
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\begin{itemize}
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\item What are Markov chains?
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@ -73,7 +73,9 @@ from the lecture on stochastic.
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\begin{subproof}
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We have $\bE[|X_n - X|] \to 0$.
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Suppose there exists an $\epsilon > 0$ such that
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$\lim_{n \to \infty} \bP[|X_n - X| > \epsilon] = c > 0$.
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$\limsup\limits_{n \to \infty} \bP[|X_n - X| > \epsilon] = c > 0$.
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W.l.o.g.~$\lim_{n \to \infty} \bP[|X_n - X| > \epsilon] = c$,
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otherwise choose an appropriate subsequence.
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We have
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\begin{IEEEeqnarray*}{rCl}
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\bE[|X_n - X|] &=& \int_\Omega |X_n - X | d\bP\\
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$X_n \xrightarrow{\bP} X \notimplies X_n\xrightarrow{L^1} X$
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\end{claim}
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\begin{subproof}
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Take $([0,1], \cB([0,1 ]), \lambda)([0,1], \cB([0,1 ]), \lambda)$
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Take $([0,1], \cB([0,1 ]), \lambda)$
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and define $X_n \coloneqq n \One_{[0, \frac{1}{n}]}$.
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We have $\bP[|X_n| > \epsilon] = \frac{1}{n}$
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for $n$ large enough.
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