some changes

inputs/a_1_counterexamples.tex

@@ -1,2 +1,6 @@
 \section{Counterexamples}
 
+Exercise 4.3
+10.2
+
+

inputs/intro.tex

@@ -9,12 +9,13 @@ in the summer term 2023 at the University Münster.
 \end{warning}
 
 These notes contain errors almost surely.
-If you find some of them or want to improve something, please send me a message:
+If you find some of them or want to improve something,
+please send me a message:\\
 \texttt{notes\_probability\_theory@jrpie.de}.
 
 
 
-Topics of this lecture:
+\paragraph{Topics of this lecture}
 \begin{enumerate}[(1)]
     \item Limit theorems: Laws of large numbers and the central limit theorem for i.i.d.~sequences,
     \item Conditional expectation and conditional probabilities,

inputs/lecture_1.tex

@@ -62,6 +62,7 @@ The converse to this fact is also true:
 \begin{proof}
     See theorem 2.4.3 in Stochastik.
 \end{proof}
+
 \begin{example}[Some important probability distribution functions]\hfill
     \begin{enumerate}[(1)]
         \item \vocab{Uniform distribution} on $[0,1]$:

inputs/lecture_10.tex

@@ -1,4 +1,4 @@
-% lecture 10 - 2023-05-09
+\lecture{10}{2023-05-09}{}
 
 % RECAP
 

inputs/lecture_11.tex

@@ -1,4 +1,5 @@
-\subsection{The central limit theorem}
+\lecture{11}{}{Intuition for the CLT}
+\subsection{The Central Limit Theorem}
 
 For $X_1, X_2,\ldots$ i.i.d.~we were looking
 at $S_n \coloneqq  \sum_{i=1}^n X_i$.

inputs/lecture_12.tex

@@ -1,4 +1,4 @@
-\lecture{12}{2023-05-16}{}
+\lecture{12}{2023-05-16}{Proof of the CLT}
 
 We now want to prove \autoref{clt}.
 The plan is to do the following:

inputs/lecture_13.tex

@@ -47,6 +47,18 @@ in this lecture. However, they are quite important.
 
 We will now sketch the proof of \autoref{levycontinuity},
 details can be found in the notes.\notes
+\begin{definition}
+    Let $(X_n)_n$ be a sequence of random variables.
+    The distribution of $(X_n)_n$ is called
+    \vocab[Distribution!tight]{tight}  (dt. ``straff''),
+    if
+    \[
+        \lim_{a \to \infty} \sup_{n \in \N} \bP[|X_n| > a] = 0.
+    \] 
+\end{definition}
+\begin{example}+[Exercise 8.1]
+    \todo{Copy}
+\end{example}
 A generalized version of \autoref{levycontinuity} is the following:
 \begin{theorem}[A generalized version of Levy's continuity \autoref{levycontinuity}]
     \label{genlevycontinuity}
@@ -55,14 +67,14 @@ A generalized version of \autoref{levycontinuity} is the following:
     for some function $\phi$ on $\R$.
     Then the following are equivalent:
     \begin{enumerate}[(a)]
-        \item The distribution of $X_n$ is \vocab[Distribution!tight]{tight} (dt. ``straff''),
+        \item The distribution of $X_n$ is tight.
-            i.e.~$\lim_{a \to \infty} \sup_{n \in \N} \bP[|X_n| > a] = 0$.
         \item $X_n \xrightarrow{(d)} X$ for some real-valued random variable $X$.
         \item  $\phi$ is the characteristic function of $X$.
         \item $\phi$ is continuous on all of $\R$.
         \item $\phi$ is continuous at $0$.
     \end{enumerate}
 \end{theorem}
+\todo{Proof of \autoref{genlevycontinuity} (Exercise 8.2)}
 \begin{example}
     Let $Z \sim \cN(0,1)$ and $X_n \coloneqq  n Z$.
     We have $\phi_{X_n}(t) = \bE[[e^{\i t X_n}] = e^{-\frac{1}{2} t^2 n^2} \xrightarrow{n \to \infty} \One_{\{t = 0\} }$.

inputs/lecture_14.tex

@@ -1,6 +1,6 @@
 \lecture{14}{2023-05-25}{Conditional expectation}
 
-\section{Conditional expectation}
+\section{Conditional Expectation}
 
 \subsection{Introduction}
 
@@ -87,7 +87,7 @@ We now want to generalize this to arbitrary random variables.
     \]
 \end{definition}
 
-\subsection{Existence of conditional probability}
+\subsection{Existence of Conditional Probability}
 
 We will give two different proves of \autoref{conditionalexpectation}.
 The first one will use orthogonal projections.

inputs/lecture_15.tex

@@ -1,9 +1,9 @@
 \lecture{15}{2023-06-06}{}
-\subsection{Properties of conditional expectation}
+\subsection{Properties of Conditional Expectation}
 
 We want to derive some properties of conditional expectation.
 
-\begin{theorem}[Law of total expectation] % Thm 1
+\begin{theorem}[Law of total expectation]
     \label{ceprop1}
     \label{totalexpectation}
     \[
@@ -50,7 +50,6 @@ We want to derive some properties of conditional expectation.
 
 \begin{theorem}[Positivity]
     \label{ceprop4}
-    % 4
     \label{cpositivity}
     If $X \ge  0$, then $\bE[X | \cG] \ge 0$ a.s.
 \end{theorem}
@@ -66,12 +65,10 @@ We want to derive some properties of conditional expectation.
 \end{proof}
 \begin{theorem}[Conditional monotone convergence theorem]
     \label{ceprop5}
-    % 5
     \label{mcmt}
     Let $X_n,X \in L^1(\Omega, \cF, \bP)$.
     Suppose $X_n \ge 0$ with $X_n \uparrow X$.
     Then $\bE[X_n|\cG] \uparrow \bE[X|\cG]$.
-    
 \end{theorem}
 \begin{proof}
     Let $Z_n$ be a version of $\bE[X_n | Y]$.
@@ -187,12 +184,10 @@ Recall
     \] 
 \end{theorem}
 \begin{proof}
-    Similar to the proof of Hölder's inequality.
     \todo{Exercise}
 \end{proof}
 
 \begin{theorem}[Tower property]
-    % 10
     \label{ceprop10}
     \label{cetower}
     Suppose $\cF \supset \cG \supset \cH$ are sub-$\sigma$-algebras.
@@ -202,11 +197,17 @@ Recall
     \] 
 \end{theorem}
 \begin{proof}
-    \todo{Exercise}
+    By definition, $\bE[\bE[X | \cG] | \cH]$ is  $\cH$-measurable.
+    For any $H \in \cH$, we have
+    \begin{IEEEeqnarray*}{rCl}
+        \int_H \bE[\bE[X | \cG] | \cH] \dif \bP
+        &=& \int_{H} \bE[X | \cG] \dif \bP\\
+        &=& \int_H X \dif \bP.
+    \end{IEEEeqnarray*}
+    Hence $\bE[\bE[X | \cG] | \cH] \overset{\text{a.s.}}{=} \bE[X | \cH]$.
 \end{proof}
 
 \begin{theorem}[Taking out what is known]
-    % 11
     \label{ceprop11}
     \label{takingoutwhatisknown}
 

inputs/lecture_16.tex

@@ -32,7 +32,7 @@
 \end{refproof}
 
 
-\subsection{The Radon Nikodym theorem}
+\subsection{The Radon Nikodym Theorem}
 
 First, let us recall some basic facts:
 \begin{fact}

inputs/lecture_17.tex

@@ -1,6 +1,6 @@
 \lecture{17}{2023-06-15}{}
 
-\subsection{Doob's martingale convergence theorem}
+\subsection{Doob's Martingale Convergence Theorem}
 
 
 \begin{definition}[Stochastic process]
@@ -37,7 +37,7 @@
     Then $(Y_n)_{n \ge 1}$ is also a (sub/super-) martingale.
 \end{lemma}
 \begin{proof}
-    Exercise. \todo{Copy}
+    Exercise. \todo{Copy Exercise 10.4}
 \end{proof}
 \begin{remark}
     The assumption of $K_n$ being constant can be weakened to

inputs/lecture_18.tex

@@ -13,7 +13,7 @@ Hence the same holds for submartingales, i.e.
     a.s.~to a finite limit, which is a.s.~finite.
 \end{lemma}
 
-\subsection{Doob's $L^p$ inequality}
+\subsection{Doob's $L^p$ Inequality}
 
 
 \begin{question}

inputs/lecture_19.tex

@@ -1,6 +1,6 @@
 \lecture{19}{2023-06-22}{}
 
-\subsection{Uniform integrability}
+\subsection{Uniform Integrability}
 
 \begin{example}
     Let $\Omega = [0,1]$, $\cF = \cB$
@@ -198,7 +198,7 @@ However, some subsets can be easily described, e.g.
     \] 
 \end{proof}
 
-\subsection{Martingale convergence theorems in $L^p, p \ge 1$}
+\subsection{Martingale Convergence Theorems in \texorpdfstring{$L^p, p \ge 1$}{$Lp, p >= 1$}}
 
 Let $(\Omega, \cF, \bP)$ as always and let $(\cF_n)_n$ always be a filtration.
 

inputs/lecture_2.tex

@@ -1,5 +1,5 @@
 \lecture{2}{}{}
-\section{Independence and product measures}
+\section{Independence and Product Measures}
 
 In order to define the notion of independence, we first need to construct
 product measures.

inputs/lecture_20.tex

@@ -66,7 +66,7 @@
 
     Hence
     \[
-        \|X_n - X\|_{L^p} % 
+        \|X_n - X\|_{L^p} %
         \le \|X_n - X_n'\|_{L^p} + \|X_n' - X'\|_{L^p} +  \|X - X'\|_{L^p} %
         \le 3 \epsilon.
     \]
@@ -118,7 +118,7 @@ we need the following theorem, which we won't prove here:
     we get the convergence.
 \end{refproof}
 
-\subsection{Stopping times}
+\subsection{Stopping Times}
 
 \begin{definition}[Stopping time]
     A random variable $T: \Omega \to \N_0 \cup  \{\infty\}$ on a filtered probability space $(\Omega, \cF, \{\cF_n\}_n, \bP)$ is called a \vocab{stopping time},
@@ -128,7 +128,6 @@ we need the following theorem, which we won't prove here:
     \]
     for all $n \in \N$.
     Equivalently, $\{T = n\}  \in \cF_n$ for all $n \in \N$.
-
 \end{definition}
 
 \begin{example}
@@ -152,7 +151,6 @@ we need the following theorem, which we won't prove here:
     T \coloneqq  \sup \{n \in \N : X_n \in  A\}
     \]
     is not a stopping time.
-
 \end{example}
 
 
@@ -167,7 +165,7 @@ we need the following theorem, which we won't prove here:
     is a stopping time.
 \end{example}
 
-\begin{example}
+\begin{fact}
     If $T_1, T_2$ are stopping times with respect to the same filtration,
     then
     \begin{itemize}
@@ -176,37 +174,32 @@ we need the following theorem, which we won't prove here:
         \item $\max \{T_1, T_2\}$
     \end{itemize}
     are stopping times.
-
+\end{fact}
+\begin{warning}
     Note that $T_1 - T_2$ is not a stopping time.
-
+\end{warning}
-\end{example}
 
 \begin{remark}
-    There are two ways to interpret the interaction between a stopping time $T$
+    There are two ways to look at the interaction between a stopping time $T$
-    and a stochastic process $(X_n)_n$.
+    and a stochastic process $(X_n)_n$:
     \begin{itemize}
-        \item The behaviour of $ X_n$ until $T$,
+        \item The behaviour of $ X_n$ until $T$, i.e.
-            i.e.~looking at the \vocab{stopped process}
             \[
             X^T \coloneqq  \left(X_{T \wedge n}\right)_{n \in \N}
-            \].
+            \]
+            is called the \vocab{stopped process}.
         \item  The value of $(X_n)_n)$ at time $T$,
             i.e.~looking at $X_T$.
     \end{itemize}
 \end{remark}
 \begin{example}
     If we look at a process
-    \[
+    \[ S_n = \sum_{i=1}^{n}  X_i \]
-    S_n = \sum_{i=1}^{n}  X_i
+    for some $(X_n)_n$,
-    \]
+    then
-    for some $(X_n)_n$, then
+    \[ S^T = (\sum_{i=1}^{T \wedge n} X_i)_n \]
-    \[
-    S^T = (\sum_{i=1}^{T \wedge n} X_i)_n
-    \]
     and
-    \[
+    \[ S_T = \sum_{i=1}^{T}  X_i. \]
-    S_T = \sum_{i=1}^{T}  X_i.
-    \]
 \end{example}
 
 \begin{theorem}
@@ -242,7 +235,6 @@ we need the following theorem, which we won't prove here:
             = 0 \text{ if $(X_n)_n$ is a martingale}.
         \end{cases}
     \end{IEEEeqnarray*}
-
 \end{proof}
 
 \begin{remark}
@@ -256,7 +248,6 @@ we need the following theorem, which we won't prove here:
         = \bE[X_0] & \text{ martingale}.
     \end{cases}
     \]
-
     However if $T$ is not bounded, this does not hold in general.
 \end{remark}
 \begin{example}
@@ -291,7 +282,7 @@ we need the following theorem, which we won't prove here:
     $\bE[X_T] = \bE[X_0]$.
 \end{theorem}
 \begin{proof}
-    (i) was dealt with in \autoref{roptionalstoppingi}.
+    (i) was already done in \autoref{roptionalstoppingi}.
 
     (ii): Since $(X_n)_n$ is bounded, we get that
     \begin{IEEEeqnarray*}{rCl}
@@ -312,7 +303,6 @@ we need the following theorem, which we won't prove here:
     \end{IEEEeqnarray*}
     Thus, we can apply (ii).
 
-
     The statement about martingales follows from
     applying this to $(X_n)_n$ and $(-X_n)_n$,
     which are both supermartingales.

inputs/lecture_22.tex

@@ -1,4 +1,4 @@
-\lecture{22}{2023-07-04}{Intro Markov Chains II}
+\lecture{22}{2023-07-04}{Introduction Markov Chains II}
 \begin{goal}
     We want to start with the basics of the theory of Markov chains.
 \end{goal}

inputs/lecture_5.tex

@@ -1,5 +1,5 @@
 \lecture{5}{2023-04-21}{}
-\subsection{The laws of large numbers}
+\subsection{The Laws of Large Numbers}
 
 
 We want to show laws of large numbers:

inputs/lecture_6.tex

@@ -1,3 +1,4 @@
+\lecture{6}{}{}
 \todo{Large parts of lecture 6 are missing}
 \begin{refproof}{lln}
     We want to deduce the SLLN (\autoref{lln}) from \autoref{thm2}.

inputs/lecture_7.tex

@@ -1,9 +1,10 @@
-% TODO \begin{goal}
+\lecture{7}{}{Kolmogorov's three series theorem}
-% TODO     We want to drop our assumptions on finite mean or variance
+\begin{goal}
-% TODO     and say something about the behaviour of $ \sum_{n \ge 1}  X_n$
+    We want to drop our assumptions on finite mean or variance
-% TODO     when the $X_n$ are independent.
+    and say something about the behaviour of $ \sum_{n \ge 1}  X_n$
-% TODO \end{goal}
+    when the $X_n$ are independent.
-\begin{theorem}[Theorem 3, Kolmogorov's three-series theorem] % Theorem 3
+\end{goal}
+\begin{theorem}[Kolmogorov's three-series theorem] % Theorem 3
     \label{thm3}
     Let $X_n$ be a family of independent random variables.
     \begin{enumerate}[(a)]
@@ -20,7 +21,7 @@
     \end{enumerate}
 \end{theorem}
 For the proof we'll need a slight generalization of \autoref{thm2}:
-\begin{theorem}[Theorem 4] % Theorem 4
+\begin{theorem} %[Theorem 4]
     \label{thm4}
     Let $\{X_n\}_n$ be independent and \vocab{uniformly bounded}
     (i.e. $\exists M < \infty : \sup_n \sup_\omega |X_n(\omega)| \le M$).
@@ -166,14 +167,13 @@ More formally:
     However
     \[
     \sum_{n}  X_n \frac{1}{n^{\frac{1}{2} + \epsilon}}
-    \] 
+    \]
     where $\bP[X_n = 1] = \bP[X_n = -1] = \frac{1}{2}$
     converges almost surely for all $\epsilon > 0$.
     And
     \[
     \sum_{n} X_n \frac{1}{n^{\frac{1}{2} - \epsilon}}
-    \] 
+    \]
     does not converge.
 
-    
 \end{example}

inputs/lecture_8.tex

@@ -24,7 +24,7 @@ of sequences of random variables.
             is again a $\sigma$-algebra, $\cT$ is indeed a $\sigma$-algebra.
         \item We have
             \[
-            \cT = \{A \in \cF ~|~ \forall i ~ \exists B \in \cB(\R)^{\otimes \N} : A = \{\omega | (X_i(\omega), X_{i+1}(\omega), \ldots) \in B\} \}. % TODO?
+            \cT = \{A \in \cF ~|~ \forall i ~ \exists B \in \cB(\R)^{\otimes \N} : A = \{\omega | (X_i(\omega), X_{i+1}(\omega), \ldots) \in B\} \}.
             \]
     \end{enumerate}
 \end{remark}
@@ -146,5 +146,3 @@ for any $k \in \N$.
     \]
     hence $\bP[T] \in \{0,1\}$.
 \end{refproof}
-
-

inputs/lecture_9.tex

@@ -1,6 +1,6 @@
+\lecture{9}{}{Percolation, Introduction to characteristic functions}
 \subsubsection{Application: Percolation}
 
-
 We will now discuss another application of Kolmogorov's $0-1$-law, percolation.
 
 \begin{definition}[\vocab{Percolation}]
@@ -41,7 +41,7 @@ For $d > 2$ this is unknown.
 We'll get back to percolation later.
 
 
-\section{Characteristic functions, weak convergence and the central limit theorem}
+\section{Characteristic Functions, Weak Convergence and the Central Limit Theorem}
 
 % Characteristic functions are also known as the \vocab{Fourier transform}.
 %Weak convergence is also known as \vocab{convergence in distribution} / \vocab{convergence in law}.
@@ -77,7 +77,7 @@ This will be the weakest notion of convergence, hence it is called
 \vocab{weak convergence}.
 This notion of convergence will be defined in terms of characteristic functions of Fourier transforms.
 
-\subsection{Characteristic functions and Fourier transform}
+\subsection{Characteristic Functions and Fourier Transform}
 
 \begin{definition}
     Consider $(\R, \cB(\R), \bP)$.
@@ -152,4 +152,3 @@ We will prove this later.
     $F(b) - F(a_n) = G(b) - G(a_n)$ hence  $F(b) = G(b)$.
     Since $F$ and $G$ are right-continuous, it follows that $F = G$.
 \end{refproof}
-

inputs/prerequisites.tex

@@ -1,7 +1,7 @@
-% This section provides a short recap of things that should be known
+This section provides a short recap of things that should be known
-% from the lecture on stochastics.
+from the lecture on stochastic.
 
-\subsection{Notions of convergence}
+\subsection{Notions of Convergence}
 \begin{definition}
     Fix a probability space $(\Omega,\cF,\bP)$.
     Let $X, X_1, X_2,\ldots$ be random variables.
@@ -147,7 +147,29 @@ The first thing that should come to mind is:
 
 We used Chebyshev's inequality. Linearity of $\bE$, $\Var(cX) = c^2\Var(X)$ and $\Var(X_1 +\ldots + X_n) = \Var(X_1) + \ldots + \Var(X_n)$ for independent $X_i$.
 
-
-
-Modes of covergence: $L^p$, in probability, a.s.
 \fi
+
+\subsection{Some Facts from Measure Theory}
+\begin{fact}+[Finite measures are {\vocab[Measure]{regular}}, Exercise 3.1]
+    Let $\mu$ be a finite measure on $(\R, \cB(\R))$.
+    Then for all $\epsilon > 0$,
+    there exists a compact set $K \in \cB(\R)$ such that
+    $\mu(K) > \mu(\R) - \epsilon$.
+\end{fact}
+\begin{proof}
+    We have $[-k,k] \uparrow \R$, hence $\mu([-k,k]) \uparrow \mu(\R)$.
+\end{proof}
+
+\begin{theorem}[Riemann-Lebesgue]
+    \label{riemann-lebesgue}
+    Let $f: \R \to  \R$ be integrable.
+    Then
+    \[
+    \lim_{n \to \infty} \int_{\R} f(x) \cos(n x) \lambda(\dif x) = 0.
+    \] 
+\end{theorem}
+
+
+
+
+

probability_theory.tex

@@ -1,4 +1,4 @@
-\documentclass[10pt,a4paper, fancyfoot, git, english]{mkessler-script}
+\documentclass[fancyfoot, git, english]{mkessler-script}
 
 \course{Probability Theory}
 \lecturer{Prof.~Chiranjib Mukherjee}
@@ -50,8 +50,10 @@
 
 \cleardoublepage
 
-%\backmatter
+\begin{landscape}
-%\chapter{Appendix}
+\section{Appendix}
+\input{inputs/a_0_distributions.tex}
+\end{landscape}
 
 \cleardoublepage
 \printvocabindex

wtheo.sty

@@ -11,6 +11,7 @@
 \usepackage[normalem]{ulem}
 \usepackage{pdflscape}
 \usepackage{longtable}
+\usepackage{colortbl}
 \usepackage{xcolor}
 \usepackage{dsfont}
 \usepackage{csquotes}
@@ -98,9 +99,15 @@
 \NewFancyTheorem[thmtools = { style = thmredmargin} , group = { big } ]{warning}
 \DeclareSimpleMathOperator{Var}
 
-\DeclareSimpleMathOperator{Bin}
+\DeclareSimpleMathOperator{Bin} % binomial distribution
-\DeclareSimpleMathOperator{Ber}
+\DeclareSimpleMathOperator{Geo} % geometric distribution
-\DeclareSimpleMathOperator{Exp}
+\DeclareSimpleMathOperator{Poi} % Poisson distribution
+
+\DeclareSimpleMathOperator{Unif} % uniform distribution
+\DeclareSimpleMathOperator{Exp} % exponential distribution
+\DeclareSimpleMathOperator{Cauchy} % Cauchy distribution
+% \DeclareSimpleMathOperator{Normal} % normal distribution
+
 
 \newcommand*\dif{\mathop{}\!\mathrm{d}}
 \newcommand\lecture[3]{\hrule{\color{darkgray}\hfill{\tiny[Lecture #1, #2]}}}