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\section{Counterexamples} \section{Counterexamples}
Exercise 4.3
10.2

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

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

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

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@ -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 For $X_1, X_2,\ldots$ i.i.d.~we were looking
at $S_n \coloneqq \sum_{i=1}^n X_i$. at $S_n \coloneqq \sum_{i=1}^n X_i$.

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@ -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}. We now want to prove \autoref{clt}.
The plan is to do the following: The plan is to do the following:

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

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

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

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

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

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

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

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

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

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

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

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

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@ -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} \label{thm3}
Let $X_n$ be a family of independent random variables. Let $X_n$ be a family of independent random variables.
\begin{enumerate}[(a)] \begin{enumerate}[(a)]
@ -20,7 +21,7 @@
\end{enumerate} \end{enumerate}
\end{theorem} \end{theorem}
For the proof we'll need a slight generalization of \autoref{thm2}: For the proof we'll need a slight generalization of \autoref{thm2}:
\begin{theorem}[Theorem 4] % Theorem 4 \begin{theorem} %[Theorem 4]
\label{thm4} \label{thm4}
Let $\{X_n\}_n$ be independent and \vocab{uniformly bounded} Let $\{X_n\}_n$ be independent and \vocab{uniformly bounded}
(i.e. $\exists M < \infty : \sup_n \sup_\omega |X_n(\omega)| \le M$). (i.e. $\exists M < \infty : \sup_n \sup_\omega |X_n(\omega)| \le M$).
@ -166,14 +167,13 @@ More formally:
However However
\[ \[
\sum_{n} X_n \frac{1}{n^{\frac{1}{2} + \epsilon}} \sum_{n} X_n \frac{1}{n^{\frac{1}{2} + \epsilon}}
\] \]
where $\bP[X_n = 1] = \bP[X_n = -1] = \frac{1}{2}$ where $\bP[X_n = 1] = \bP[X_n = -1] = \frac{1}{2}$
converges almost surely for all $\epsilon > 0$. converges almost surely for all $\epsilon > 0$.
And And
\[ \[
\sum_{n} X_n \frac{1}{n^{\frac{1}{2} - \epsilon}} \sum_{n} X_n \frac{1}{n^{\frac{1}{2} - \epsilon}}
\] \]
does not converge. does not converge.
\end{example} \end{example}

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@ -24,7 +24,7 @@ of sequences of random variables.
is again a $\sigma$-algebra, $\cT$ is indeed a $\sigma$-algebra. is again a $\sigma$-algebra, $\cT$ is indeed a $\sigma$-algebra.
\item We have \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{enumerate}
\end{remark} \end{remark}
@ -146,5 +146,3 @@ for any $k \in \N$.
\] \]
hence $\bP[T] \in \{0,1\}$. hence $\bP[T] \in \{0,1\}$.
\end{refproof} \end{refproof}

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

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@ -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} \begin{definition}
Fix a probability space $(\Omega,\cF,\bP)$. Fix a probability space $(\Omega,\cF,\bP)$.
Let $X, X_1, X_2,\ldots$ be random variables. 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 $X_i$. 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 \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}

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

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@ -11,6 +11,7 @@
\usepackage[normalem]{ulem} \usepackage[normalem]{ulem}
\usepackage{pdflscape} \usepackage{pdflscape}
\usepackage{longtable} \usepackage{longtable}
\usepackage{colortbl}
\usepackage{xcolor} \usepackage{xcolor}
\usepackage{dsfont} \usepackage{dsfont}
\usepackage{csquotes} \usepackage{csquotes}
@ -98,9 +99,15 @@
\NewFancyTheorem[thmtools = { style = thmredmargin} , group = { big } ]{warning} \NewFancyTheorem[thmtools = { style = thmredmargin} , group = { big } ]{warning}
\DeclareSimpleMathOperator{Var} \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*\dif{\mathop{}\!\mathrm{d}}
\newcommand\lecture[3]{\hrule{\color{darkgray}\hfill{\tiny[Lecture #1, #2]}}} \newcommand\lecture[3]{\hrule{\color{darkgray}\hfill{\tiny[Lecture #1, #2]}}}