some small changes

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Josia Pietsch 2023-07-07 17:42:38 +02:00
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13 changed files with 224 additions and 23 deletions

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pdf: init
latexmk
latexmk < /dev/null
clean:
latexmk -c

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\section{Counterexamples}
\section{(Counter)examples}
Consistent families and inconsistent families
Notions of convergence
Exercise 4.3
10.2
Martingales converging a.s.~but not in $L^1$.
Stopping times

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Important stuff not done in the lecture.
Moments:
$\bE[X^k]$
\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}
Laplace transforms

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\lecture{2}{}{}
\lecture{2}{2023-04-11}{Independence, Kolmogorov's consistency theorem, consistent families}
\section{Independence and Product Measures}
In order to define the notion of independence, we first need to construct
@ -80,9 +80,9 @@ to an infinite number of random variables.
\begin{theorem}[Kolmogorov extension / consistency theorem]
\label{thm:kolmogorovconsistency}
Informally:
\footnote{Informally:
``Probability measures are determined by finite-dimensional marginals
(as long as these marginals are nice)''
(as long as these marginals are nice)''}
Let $\bP_n, n \in \N$ be probability measures on $(\R^n, \cB(\R^n))$
which are \vocab{consistent},
@ -117,5 +117,3 @@ to an infinite number of random variables.
the distribution function of $X_i$.
In the case of $F_1 = \ldots = F_n$, then $X_1,\ldots, X_n$ are i.i.d.
\end{example}

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@ -154,4 +154,5 @@ we are going to use the following:
= \lambda_n(C_1) + \lambda_n(C_2)
\]
by the definition of the finite product measure.
\phantom\qedhere
\end{refproof}

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@ -1,4 +1,4 @@
\lecture{5}{2023-04-21}{}
\lecture{5}{2023-04-21}{Laws of large numbers}
\subsection{The Laws of Large Numbers}
@ -44,23 +44,14 @@ For the proof of (b) we need the following general result:
\begin{theorem}
\label{thm2}
Let $X_1, X_2, \ldots$ be independent (but not necessarily identically distributed) random variables with $\bE[X_i] = 0$ for all $i$
and $\sum_{i=1}^n \Var(X_i) < \infty$.
and
\[\sum_{i=1}^n \Var(X_i) < \infty.\]
Then $\sum_{n \ge 1} X_n$ converges almost surely.
\end{theorem}
We'll prove this later\todo{Move proof}
\begin{proof}
\end{proof}
\begin{question}
Does the converse hold? I.e.~does $\sum_{n \ge 1} X_n < \infty$ a.s.~
then $\sum_{n \ge 1} \Var(X_n) < \infty$.
\end{question}
This does not hold. Consider for example $X_n = \frac{1}{n^2} \delta_n + \frac{1}{n^2} \delta_{-n} + (1-\frac{2}{n^2}) \delta_0$.
\begin{refproof}{lln}
\begin{enumerate}
\item[(b)]
\end{enumerate}
\end{refproof}

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@ -5,6 +5,7 @@
when the $X_n$ are independent.
\end{goal}
\begin{theorem}[Kolmogorov's three-series theorem] % Theorem 3
\label{thm:kolmogorovthreeseries}
\label{thm3}
Let $X_n$ be a family of independent random variables.
\begin{enumerate}[(a)]
@ -43,7 +44,7 @@ For the proof we'll need a slight generalization of \autoref{thm2}:
almost surely.
Let $A_n \coloneqq \{\omega : |X_n(\omega)| > C\}$.
Since the first series $\sum_{n \ge 1} \bP(A_n) < \infty$,
by Borel-Cantelli, $\bP[\text{infinitely many $A_n$ occcur}] = 0$.
by Borel-Cantelli, $\bP[\text{infinitely many $A_n$ occur}] = 0$.
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
\subsection{Characteristic Functions and Fourier Transform}
\begin{definition}
\label{def:characteristicfunction}
Consider $(\R, \cB(\R), \bP)$.
The \vocab{characteristic function} of $\bP$ is defined as
\begin{IEEEeqnarray*}{rCl}

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@ -184,6 +184,7 @@ Unfortunately, we won't prove \autoref{bochnersthm} in this lecture.
\begin{definition}[Convergence in distribution / weak convergence]
\label{def:weakconvergence}
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
\[
\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.
$G_1, G_2, \ldots$ is a subsequence of $F_1, F_2,\ldots$.
However $G_1, G_2, \ldots$ is not converging to $F$,
as this would fail at $x_0$. This is a contradiction.
\end{refproof}

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@ -1,3 +1,4 @@
\pagebreak
\lecture{22}{2023-07-04}{Introduction Markov Chains II}
\section{Markov Chains}
\todo{Merge this with the end of lecture 21}

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inputs/lecture_23.tex Normal file
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\lecture{23}{2023-07-06}{}
\section{Recap}
In this lecture we will recall the most important point from the lecture.
\subsection{Construction of iid random variables.}
\begin{itemize}
\item Definition of a consistent family (\autoref{def:consistentfamily})
\item Important construction:
Consider a distribution function $F$ and define
\[
\prod_{i=1}^n (F(b_i) - F(a_i)) \text{\reflectbox{$\coloneqq$}}
\mu_n \left( (a_1,b_1] \times x \ldots \times x (a_n, b_n] \right).
\]
\item Examples of consistent and inconsistent families
\todo{Exercises}
\item Kolmogorov's consistency theorem
(\autoref{thm:kolmogorovconsistency})
\end{itemize}
\subsection{Limit theorems}
\begin{itemize}
\item Work with iid.~random variables.
\item Notions of convergence (\autoref{def:convergence})
\item Implications between different notions of convergence (very important) and counter examples.
(\autoref{thm:convergenceimplications})
\item \begin{itemize}
\item Laws of large numbers: (\autoref{lln})
\begin{itemize}
\item WLLN: convergence in probability
\item SLLN: weak convergence
\end{itemize}
\end{itemize}
\item \autoref{thm2} (building block for SLLN):
Let $(X_n)$ be independent with mean $0$ and $\sum \sigma_n^2 < \infty$,
then $ \sum X_n $ converges a.s.
\begin{itemize}
\item Counter examples showing that $\impliedby$ does not hold in general are important
\item $\impliedby$ holds for iid.~uniformly bounded random variables
\item Application:
$\sum_{i=1}^{\infty} \frac{(\pm_1)}{n^{\frac{1}{2} + \epsilon}}$ converges a.s.~for all $\epsilon > 0$.
$\sum \frac{\pm 1}{ n^{\frac{1}{2} -\epsilon}}$ does not converge a.s.~for any $\epsilon > 0$.
\end{itemize}
\item Kolmogorov's inequality (\autoref{thm:kolmogorovineq})
\item Kolmogorov's $0-1$-law. (\autoref{kolmogorov01})
In particular, a series of independent random variables converges with probability $0$ or $1$.
\item Kolmogorov's 3 series theorem. (\autoref{thm:kolmogorovthreeseries})
\begin{itemize}
\item What are those $3$ series?
\item Applications
\end{itemize}
\end{itemize}
\subsubsection{Fourier transform / characteristic functions / weak convegence}
\begin{itemize}
\item Definition of Fourier transform
(\autoref{def:characteristicfunction})
\item The Fourier transform uniquely determines the probability distribution.
It is bounded, so many theorems are easily applicable.
\item Uniqueness theorem (\autoref{charfuncuniqueness}),
inversion formula (\autoref{inversionformula}), ...
\item Levy's continuity theorem (\autoref{levycontinuity}),
(\autoref{genlevycontinuity})
\item Bockner's theorem for positive definite function % TODO REF
\item Bockner's theorem for the mass at a point (\autoref{bochnersformula})
\item Related notions
\todo{TODO}
\begin{itemize}
\item Laplace transforms $\bE[e^{-\lambda X}]$ for some $\lambda > 0$
(not done in the lecture, but still useful).
\item Moments $\bE[X^k]$ (not done in the lecture, but still useful)
All moments together uniquely determine the distribution.
\end{itemize}
\end{itemize}
\paragraph{Weak convergence}
\begin{itemize}
\item Definition of weak convergence % ( test against continuous, bounded functions).
(\autoref{def:weakconvergence})
\item Examples:
\begin{itemize}
\item $(\delta_{\frac{1}{n}})_n$,
\item $(\frac{1}{2} \delta_{-\frac{1}{n}} + \frac{1}{2} \delta_{\frac{1}{n}}$,
\item $(\cN(0, \frac{1}{n}))_n$,
\item $(\frac{1}{n} \delta_n + (1- \frac{1}{n}) \delta_{\frac{1}{n}})_n$.
\end{itemize}
\item Non-examples: $(\delta_n)_n$
\item How does one prove weak convergence? How does one write this down in a clear way?
% TODO
\end{itemize}
\paragraph{Convolution}
\begin{itemize}
\item Definition of convolution.
\todo{Copy from exercise sheet and write a section about this}
\item $X_i \sim \mu_i \text{ iid. }\implies X_1 + \ldots + X_n \sim \mu_1 \ast \ldots \ast \mu_n$.
\end{itemize}
\subsubsection{CLT}
\begin{itemize}
\item Statement of the CLT
\item Several versions:
\begin{itemize}
\item iid (\autoref{clt}),
\item Lindeberg (\autoref{lindebergclt}),
\item Luyapanov (\autoref{lyapunovclt})
\end{itemize}
\item How to apply this? Exercises!
\end{itemize}
\subsection{Conditional expectation}
\begin{itemize}
\item Definition and existence of conditional expectation for $X \in L^1(\Omega, \cF, \bP)$
\item If $H = L^2(\Omega, \cF, \bP)$, then $\bE[ \cdot | \cG]$
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 (Non-)examples of mutually absolutely continuous measures
Singularity in this context? % TODO
\end{itemize}
\subsection{Martingales}
\begin{itemize}
\item Definition of Martingales
\item Doob's convergence theorem, Upcrossing inequality
(downcrossings for submartingales)
\item Examples of Martingales converging a.s.~but not in $L^1$
\item Bounded in $L^2$ $\implies$ convergence in $L^2$.
\item Martingale increments are orthogonal in $L^2$!
\item Doob's (sub-)martingale inequalities
\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.
\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?
\item (Non-)examples of stopping times
\item \textbf{Optional stopping theorem} - be really comfortable with this.
\end{itemize}
\subsection{Markov Chains}
\begin{itemize}
\item What are Markov chains?
\item State space, initial distribution
\item Important examples
\item \textbf{What is the relation between Martingales and Markov chains?}
$u$ \vocab{harmonic} $\iff Lu = 0$.
(sub-/super-) harmonic $u$ $\iff$ for a Markov chain $(X_n)$,
$u(X_n)$ is a (sub-/super-)martingale
\item Dirichlet problem
(Not done in the lecture)
\item ... (more in Probability Theory II)
\end{itemize}

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@ -3,6 +3,7 @@ from the lecture on stochastic.
\subsection{Notions of Convergence}
\begin{definition}
\label{def:convergence}
Fix a probability space $(\Omega,\cF,\bP)$.
Let $X, X_1, X_2,\ldots$ be random variables.
\begin{itemize}
@ -30,9 +31,10 @@ from the lecture on stochastic.
\]
\end{itemize}
\end{definition}
% TODO Connect to ANaIII
% TODO Connect to AnaIII
\begin{theorem}
\label{thm:convergenceimplications}
\vspace{10pt}
Let $X$ be a random variable and $X_n, n \in \N$ a sequence of random variables.
Then
@ -80,6 +82,9 @@ from the lecture on stochastic.
&=& \epsilon \cdot c > 0 \lightning
\end{IEEEeqnarray*}
\todo{Improve this with Markov}
\todo{Counter examples}
\todo{weak convergence}
\todo{$L^p$ convergence}
\end{subproof}
\begin{claim}
$X_n \xrightarrow{\bP} X \notimplies X_n\xrightarrow{L^1} X$