some small changes
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13 changed files with 224 additions and 23 deletions
2
Makefile
2
Makefile
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@ -1,5 +1,5 @@
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pdf: init
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latexmk
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latexmk < /dev/null
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clean:
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latexmk -c
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@ -1,5 +1,17 @@
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\section{Counterexamples}
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\section{(Counter)examples}
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Consistent families and inconsistent families
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Notions of convergence
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Exercise 4.3
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10.2
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Martingales converging a.s.~but not in $L^1$.
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Stopping times
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21
inputs/a_2_additional_stuff.tex
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21
inputs/a_2_additional_stuff.tex
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@ -0,0 +1,21 @@
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Important stuff not done in the lecture.
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Moments:
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$\bE[X^k]$
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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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Laplace transforms
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@ -1,4 +1,4 @@
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\lecture{2}{}{}
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\lecture{2}{2023-04-11}{Independence, Kolmogorov's consistency theorem, consistent families}
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\section{Independence and Product Measures}
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In order to define the notion of independence, we first need to construct
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@ -80,9 +80,9 @@ to an infinite number of random variables.
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\begin{theorem}[Kolmogorov extension / consistency theorem]
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\label{thm:kolmogorovconsistency}
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Informally:
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\footnote{Informally:
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``Probability measures are determined by finite-dimensional marginals
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(as long as these marginals are nice)''
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(as long as these marginals are nice)''}
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Let $\bP_n, n \in \N$ be probability measures on $(\R^n, \cB(\R^n))$
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which are \vocab{consistent},
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@ -117,5 +117,3 @@ to an infinite number of random variables.
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the distribution function of $X_i$.
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In the case of $F_1 = \ldots = F_n$, then $X_1,\ldots, X_n$ are i.i.d.
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\end{example}
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@ -154,4 +154,5 @@ we are going to use the following:
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= \lambda_n(C_1) + \lambda_n(C_2)
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\]
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by the definition of the finite product measure.
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\phantom\qedhere
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\end{refproof}
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@ -1,4 +1,4 @@
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\lecture{5}{2023-04-21}{}
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\lecture{5}{2023-04-21}{Laws of large numbers}
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\subsection{The Laws of Large Numbers}
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@ -44,23 +44,14 @@ For the proof of (b) we need the following general result:
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\begin{theorem}
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\label{thm2}
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Let $X_1, X_2, \ldots$ be independent (but not necessarily identically distributed) random variables with $\bE[X_i] = 0$ for all $i$
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and $\sum_{i=1}^n \Var(X_i) < \infty$.
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and
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\[\sum_{i=1}^n \Var(X_i) < \infty.\]
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Then $\sum_{n \ge 1} X_n$ converges almost surely.
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\end{theorem}
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We'll prove this later\todo{Move proof}
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\begin{proof}
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\end{proof}
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\begin{question}
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Does the converse hold? I.e.~does $\sum_{n \ge 1} X_n < \infty$ a.s.~
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then $\sum_{n \ge 1} \Var(X_n) < \infty$.
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\end{question}
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This does not hold. Consider for example $X_n = \frac{1}{n^2} \delta_n + \frac{1}{n^2} \delta_{-n} + (1-\frac{2}{n^2}) \delta_0$.
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\begin{refproof}{lln}
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\begin{enumerate}
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\item[(b)]
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\end{enumerate}
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\end{refproof}
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@ -5,6 +5,7 @@
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when the $X_n$ are independent.
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\end{goal}
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\begin{theorem}[Kolmogorov's three-series theorem] % Theorem 3
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\label{thm:kolmogorovthreeseries}
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\label{thm3}
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Let $X_n$ be a family of independent random variables.
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\begin{enumerate}[(a)]
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@ -43,7 +44,7 @@ For the proof we'll need a slight generalization of \autoref{thm2}:
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almost surely.
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Let $A_n \coloneqq \{\omega : |X_n(\omega)| > C\}$.
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Since the first series $\sum_{n \ge 1} \bP(A_n) < \infty$,
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by Borel-Cantelli, $\bP[\text{infinitely many $A_n$ occcur}] = 0$.
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by Borel-Cantelli, $\bP[\text{infinitely many $A_n$ occur}] = 0$.
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For the proof of (b), suppose $\sum_{n\ge 1} X_n(\omega) < \infty$
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@ -80,6 +80,7 @@ This notion of convergence will be defined in terms of characteristic functions
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\subsection{Characteristic Functions and Fourier Transform}
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\begin{definition}
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\label{def:characteristicfunction}
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Consider $(\R, \cB(\R), \bP)$.
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The \vocab{characteristic function} of $\bP$ is defined as
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\begin{IEEEeqnarray*}{rCl}
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@ -184,6 +184,7 @@ Unfortunately, we won't prove \autoref{bochnersthm} in this lecture.
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\begin{definition}[Convergence in distribution / weak convergence]
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\label{def:weakconvergence}
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We say that $\bP_n \subseteq M_1(\R)$ \vocab[Convergence!weak]{converges weakly} towards $\bP \in M_1(\R)$ (notation: $\bP_n \implies \bP$), iff
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\[
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\forall f \in C_b(\R)~ \int f d\bP_n \to \int f d\bP.
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@ -264,7 +264,6 @@ which converges.
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$G_1, G_2, \ldots$ is a subsequence of $F_1, F_2,\ldots$.
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However $G_1, G_2, \ldots$ is not converging to $F$,
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as this would fail at $x_0$. This is a contradiction.
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\end{refproof}
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@ -1,3 +1,4 @@
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\pagebreak
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\lecture{22}{2023-07-04}{Introduction Markov Chains II}
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\section{Markov Chains}
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\todo{Merge this with the end of lecture 21}
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170
inputs/lecture_23.tex
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170
inputs/lecture_23.tex
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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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\subsection{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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\item Important construction:
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Consider a distribution function $F$ and define
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\[
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\prod_{i=1}^n (F(b_i) - F(a_i)) \text{\reflectbox{$\coloneqq$}}
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\mu_n \left( (a_1,b_1] \times x \ldots \times x (a_n, b_n] \right).
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\]
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\item Examples of consistent and inconsistent families
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\todo{Exercises}
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\item Kolmogorov's consistency theorem
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(\autoref{thm:kolmogorovconsistency})
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\end{itemize}
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\subsection{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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\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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\begin{itemize}
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\item Counter examples showing that $\impliedby$ does not hold in general are important
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\item $\impliedby$ holds for iid.~uniformly bounded random variables
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\item Application:
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$\sum_{i=1}^{\infty} \frac{(\pm_1)}{n^{\frac{1}{2} + \epsilon}}$ converges a.s.~for all $\epsilon > 0$.
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$\sum \frac{\pm 1}{ n^{\frac{1}{2} -\epsilon}}$ does not converge a.s.~for any $\epsilon > 0$.
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\end{itemize}
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\item Kolmogorov's inequality (\autoref{thm:kolmogorovineq})
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\item Kolmogorov's $0-1$-law. (\autoref{kolmogorov01})
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In particular, a series of independent random variables converges with probability $0$ or $1$.
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\item Kolmogorov's 3 series theorem. (\autoref{thm:kolmogorovthreeseries})
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\begin{itemize}
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\item What are those $3$ series?
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\item Applications
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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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\begin{itemize}
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\item Definition of Fourier transform
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(\autoref{def:characteristicfunction})
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\item The Fourier transform uniquely determines the probability distribution.
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It is bounded, so many theorems are easily applicable.
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\item Uniqueness theorem (\autoref{charfuncuniqueness}),
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inversion formula (\autoref{inversionformula}), ...
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\item Levy's continuity theorem (\autoref{levycontinuity}),
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(\autoref{genlevycontinuity})
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\item Bockner's theorem for positive definite function % TODO REF
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\item Bockner's theorem for the mass at a point (\autoref{bochnersformula})
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\item Related notions
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\todo{TODO}
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\begin{itemize}
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\item Laplace transforms $\bE[e^{-\lambda X}]$ for some $\lambda > 0$
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(not done in the lecture, but still useful).
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\item Moments $\bE[X^k]$ (not done in the lecture, but still useful)
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All moments together uniquely determine the distribution.
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\end{itemize}
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\end{itemize}
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\paragraph{Weak convergence}
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\begin{itemize}
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\item Definition of weak convergence % ( test against continuous, bounded functions).
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(\autoref{def:weakconvergence})
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\item Examples:
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\begin{itemize}
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\item $(\delta_{\frac{1}{n}})_n$,
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\item $(\frac{1}{2} \delta_{-\frac{1}{n}} + \frac{1}{2} \delta_{\frac{1}{n}}$,
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\item $(\cN(0, \frac{1}{n}))_n$,
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\item $(\frac{1}{n} \delta_n + (1- \frac{1}{n}) \delta_{\frac{1}{n}})_n$.
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\end{itemize}
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\item Non-examples: $(\delta_n)_n$
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\item How does one prove weak convergence? How does one write this down in a clear way?
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% TODO
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\end{itemize}
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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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\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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\begin{itemize}
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\item Statement of the CLT
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\item Several versions:
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\begin{itemize}
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\item iid (\autoref{clt}),
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\item Lindeberg (\autoref{lindebergclt}),
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\item Luyapanov (\autoref{lyapunovclt})
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\end{itemize}
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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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\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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is the (unique) projection on the closed subspace $L^2(\Omega, \cG, \bP)$.
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Why is this a closed subspace? Why is the projection orthogonal?
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\item Radon-Nikodym Theorem (Proof not relevant for the exam)
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\item (Non-)examples of mutually absolutely continuous measures
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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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\begin{itemize}
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\item Definition of Martingales
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\item Doob's convergence theorem, Upcrossing inequality
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(downcrossings for submartingales)
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\item Examples of Martingales converging a.s.~but not in $L^1$
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\item Bounded in $L^2$ $\implies$ convergence in $L^2$.
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\item Martingale increments are orthogonal in $L^2$!
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\item Doob's (sub-)martingale inequalities
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\item $\bP[\sup_{k \le n} M_k \ge x]$ $\leadsto$ Look at martingale inequalities!
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Estimates might come from Doob's inequalities if $(M_k)_k$ is a (sub-)martingale.
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\item Doob's $L^p$ convergence theorem.
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\begin{itemize}
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\item Why is $p > 1$ important? \textbf{Role of Banach-Alaoglu}
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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 (Non-)examples of stopping times
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\item \textbf{Optional stopping theorem} - be really comfortable with this.
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\end{itemize}
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\subsection{Markov Chains}
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\begin{itemize}
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\item What are Markov chains?
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\item State space, initial distribution
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\item Important examples
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\item \textbf{What is the relation between Martingales and Markov chains?}
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$u$ \vocab{harmonic} $\iff Lu = 0$.
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(sub-/super-) harmonic $u$ $\iff$ for a Markov chain $(X_n)$,
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$u(X_n)$ is a (sub-/super-)martingale
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\item Dirichlet problem
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(Not done in the lecture)
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\item ... (more in Probability Theory II)
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\end{itemize}
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@ -3,6 +3,7 @@ from the lecture on stochastic.
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\subsection{Notions of Convergence}
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\begin{definition}
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\label{def:convergence}
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Fix a probability space $(\Omega,\cF,\bP)$.
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Let $X, X_1, X_2,\ldots$ be random variables.
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\begin{itemize}
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\]
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\end{itemize}
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\end{definition}
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% TODO Connect to ANaIII
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% TODO Connect to AnaIII
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\begin{theorem}
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\label{thm:convergenceimplications}
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\vspace{10pt}
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Let $X$ be a random variable and $X_n, n \in \N$ a sequence of random variables.
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Then
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&=& \epsilon \cdot c > 0 \lightning
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\end{IEEEeqnarray*}
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\todo{Improve this with Markov}
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\todo{Counter examples}
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\todo{weak convergence}
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\todo{$L^p$ convergence}
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\end{subproof}
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\begin{claim}
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$X_n \xrightarrow{\bP} X \notimplies X_n\xrightarrow{L^1} X$
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