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@ -1,2 +1,6 @@
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\section{Counterexamples}
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Exercise 4.3
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10.2
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@ -9,12 +9,13 @@ in the summer term 2023 at the University Münster.
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\end{warning}
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These notes contain errors almost surely.
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If you find some of them or want to improve something, please send me a message:
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If you find some of them or want to improve something,
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please send me a message:\\
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\texttt{notes\_probability\_theory@jrpie.de}.
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Topics of this lecture:
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\paragraph{Topics of this lecture}
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\begin{enumerate}[(1)]
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\item Limit theorems: Laws of large numbers and the central limit theorem for i.i.d.~sequences,
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\item Conditional expectation and conditional probabilities,
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@ -62,6 +62,7 @@ The converse to this fact is also true:
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\begin{proof}
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See theorem 2.4.3 in Stochastik.
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\end{proof}
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\begin{example}[Some important probability distribution functions]\hfill
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\begin{enumerate}[(1)]
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\item \vocab{Uniform distribution} on $[0,1]$:
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@ -1,4 +1,4 @@
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% lecture 10 - 2023-05-09
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\lecture{10}{2023-05-09}{}
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% RECAP
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@ -1,4 +1,5 @@
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\subsection{The central limit theorem}
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\lecture{11}{}{Intuition for the CLT}
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\subsection{The Central Limit Theorem}
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For $X_1, X_2,\ldots$ i.i.d.~we were looking
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at $S_n \coloneqq \sum_{i=1}^n X_i$.
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@ -1,4 +1,4 @@
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\lecture{12}{2023-05-16}{}
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\lecture{12}{2023-05-16}{Proof of the CLT}
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We now want to prove \autoref{clt}.
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The plan is to do the following:
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@ -47,6 +47,18 @@ in this lecture. However, they are quite important.
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We will now sketch the proof of \autoref{levycontinuity},
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details can be found in the notes.\notes
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\begin{definition}
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Let $(X_n)_n$ be a sequence of random variables.
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The distribution of $(X_n)_n$ is called
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\vocab[Distribution!tight]{tight} (dt. ``straff''),
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if
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\[
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\lim_{a \to \infty} \sup_{n \in \N} \bP[|X_n| > a] = 0.
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\]
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\end{definition}
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\begin{example}+[Exercise 8.1]
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\todo{Copy}
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\end{example}
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A generalized version of \autoref{levycontinuity} is the following:
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\begin{theorem}[A generalized version of Levy's continuity \autoref{levycontinuity}]
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\label{genlevycontinuity}
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@ -55,14 +67,14 @@ A generalized version of \autoref{levycontinuity} is the following:
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for some function $\phi$ on $\R$.
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Then the following are equivalent:
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\begin{enumerate}[(a)]
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\item The distribution of $X_n$ is \vocab[Distribution!tight]{tight} (dt. ``straff''),
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i.e.~$\lim_{a \to \infty} \sup_{n \in \N} \bP[|X_n| > a] = 0$.
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\item The distribution of $X_n$ is tight.
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\item $X_n \xrightarrow{(d)} X$ for some real-valued random variable $X$.
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\item $\phi$ is the characteristic function of $X$.
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\item $\phi$ is continuous on all of $\R$.
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\item $\phi$ is continuous at $0$.
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\end{enumerate}
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\end{theorem}
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\todo{Proof of \autoref{genlevycontinuity} (Exercise 8.2)}
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\begin{example}
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Let $Z \sim \cN(0,1)$ and $X_n \coloneqq n Z$.
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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 @@
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\lecture{14}{2023-05-25}{Conditional expectation}
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\section{Conditional expectation}
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\section{Conditional Expectation}
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\subsection{Introduction}
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@ -87,7 +87,7 @@ We now want to generalize this to arbitrary random variables.
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\]
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\end{definition}
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\subsection{Existence of conditional probability}
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\subsection{Existence of Conditional Probability}
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We will give two different proves of \autoref{conditionalexpectation}.
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The first one will use orthogonal projections.
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@ -1,9 +1,9 @@
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\lecture{15}{2023-06-06}{}
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\subsection{Properties of conditional expectation}
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\subsection{Properties of Conditional Expectation}
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We want to derive some properties of conditional expectation.
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\begin{theorem}[Law of total expectation] % Thm 1
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\begin{theorem}[Law of total expectation]
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\label{ceprop1}
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\label{totalexpectation}
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\[
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\begin{theorem}[Positivity]
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\label{ceprop4}
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% 4
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\label{cpositivity}
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If $X \ge 0$, then $\bE[X | \cG] \ge 0$ a.s.
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\end{theorem}
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\end{proof}
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\begin{theorem}[Conditional monotone convergence theorem]
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\label{ceprop5}
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% 5
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\label{mcmt}
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Let $X_n,X \in L^1(\Omega, \cF, \bP)$.
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Suppose $X_n \ge 0$ with $X_n \uparrow X$.
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Then $\bE[X_n|\cG] \uparrow \bE[X|\cG]$.
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\end{theorem}
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\begin{proof}
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Let $Z_n$ be a version of $\bE[X_n | Y]$.
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@ -187,12 +184,10 @@ Recall
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\]
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\end{theorem}
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\begin{proof}
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Similar to the proof of Hölder's inequality.
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\todo{Exercise}
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\end{proof}
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\begin{theorem}[Tower property]
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% 10
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\label{ceprop10}
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\label{cetower}
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Suppose $\cF \supset \cG \supset \cH$ are sub-$\sigma$-algebras.
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@ -202,11 +197,17 @@ Recall
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\]
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\end{theorem}
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\begin{proof}
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\todo{Exercise}
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By definition, $\bE[\bE[X | \cG] | \cH]$ is $\cH$-measurable.
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For any $H \in \cH$, we have
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\begin{IEEEeqnarray*}{rCl}
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\int_H \bE[\bE[X | \cG] | \cH] \dif \bP
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&=& \int_{H} \bE[X | \cG] \dif \bP\\
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&=& \int_H X \dif \bP.
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\end{IEEEeqnarray*}
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Hence $\bE[\bE[X | \cG] | \cH] \overset{\text{a.s.}}{=} \bE[X | \cH]$.
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\end{proof}
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\begin{theorem}[Taking out what is known]
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% 11
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\label{ceprop11}
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\label{takingoutwhatisknown}
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@ -32,7 +32,7 @@
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\end{refproof}
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\subsection{The Radon Nikodym theorem}
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\subsection{The Radon Nikodym Theorem}
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First, let us recall some basic facts:
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\begin{fact}
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@ -1,6 +1,6 @@
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\lecture{17}{2023-06-15}{}
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\subsection{Doob's martingale convergence theorem}
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\subsection{Doob's Martingale Convergence Theorem}
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\begin{definition}[Stochastic process]
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Then $(Y_n)_{n \ge 1}$ is also a (sub/super-) martingale.
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\end{lemma}
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\begin{proof}
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Exercise. \todo{Copy}
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Exercise. \todo{Copy Exercise 10.4}
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\end{proof}
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\begin{remark}
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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.
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a.s.~to a finite limit, which is a.s.~finite.
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\end{lemma}
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\subsection{Doob's $L^p$ inequality}
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\subsection{Doob's $L^p$ Inequality}
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\begin{question}
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@ -1,6 +1,6 @@
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\lecture{19}{2023-06-22}{}
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\subsection{Uniform integrability}
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\subsection{Uniform Integrability}
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\begin{example}
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Let $\Omega = [0,1]$, $\cF = \cB$
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\]
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\end{proof}
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\subsection{Martingale convergence theorems in $L^p, p \ge 1$}
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\subsection{Martingale Convergence Theorems in \texorpdfstring{$L^p, p \ge 1$}{$Lp, p >= 1$}}
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Let $(\Omega, \cF, \bP)$ as always and let $(\cF_n)_n$ always be a filtration.
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@ -1,5 +1,5 @@
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\lecture{2}{}{}
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\section{Independence and product measures}
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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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product measures.
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@ -66,7 +66,7 @@
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Hence
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\[
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\|X_n - X\|_{L^p} %
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\|X_n - X\|_{L^p} %
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\le \|X_n - X_n'\|_{L^p} + \|X_n' - X'\|_{L^p} + \|X - X'\|_{L^p} %
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\le 3 \epsilon.
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\]
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@ -118,7 +118,7 @@ we need the following theorem, which we won't prove here:
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we get the convergence.
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\end{refproof}
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\subsection{Stopping times}
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\subsection{Stopping Times}
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\begin{definition}[Stopping time]
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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},
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\]
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for all $n \in \N$.
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Equivalently, $\{T = n\} \in \cF_n$ for all $n \in \N$.
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\end{definition}
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\begin{example}
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T \coloneqq \sup \{n \in \N : X_n \in A\}
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\]
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is not a stopping time.
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\end{example}
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@ -167,7 +165,7 @@ we need the following theorem, which we won't prove here:
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is a stopping time.
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\end{example}
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\begin{example}
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\begin{fact}
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If $T_1, T_2$ are stopping times with respect to the same filtration,
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then
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\begin{itemize}
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\item $\max \{T_1, T_2\}$
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\end{itemize}
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are stopping times.
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\end{fact}
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\begin{warning}
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Note that $T_1 - T_2$ is not a stopping time.
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\end{example}
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\end{warning}
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\begin{remark}
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There are two ways to interpret the interaction between a stopping time $T$
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and a stochastic process $(X_n)_n$.
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There are two ways to look at the interaction between a stopping time $T$
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and a stochastic process $(X_n)_n$:
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\begin{itemize}
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\item The behaviour of $ X_n$ until $T$,
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i.e.~looking at the \vocab{stopped process}
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\item The behaviour of $ X_n$ until $T$, i.e.
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\[
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X^T \coloneqq \left(X_{T \wedge n}\right)_{n \in \N}
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\].
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\]
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is called the \vocab{stopped process}.
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\item The value of $(X_n)_n)$ at time $T$,
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i.e.~looking at $X_T$.
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\end{itemize}
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\end{remark}
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\begin{example}
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If we look at a process
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\[
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S_n = \sum_{i=1}^{n} X_i
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\]
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for some $(X_n)_n$, then
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\[
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S^T = (\sum_{i=1}^{T \wedge n} X_i)_n
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\]
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\[ S_n = \sum_{i=1}^{n} X_i \]
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for some $(X_n)_n$,
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then
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\[ S^T = (\sum_{i=1}^{T \wedge n} X_i)_n \]
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and
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\[
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S_T = \sum_{i=1}^{T} X_i.
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\]
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\[ S_T = \sum_{i=1}^{T} X_i. \]
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\end{example}
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\begin{theorem}
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= 0 \text{ if $(X_n)_n$ is a martingale}.
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\end{cases}
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\end{IEEEeqnarray*}
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\end{proof}
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\begin{remark}
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= \bE[X_0] & \text{ martingale}.
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\end{cases}
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\]
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However if $T$ is not bounded, this does not hold in general.
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\end{remark}
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\begin{example}
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$\bE[X_T] = \bE[X_0]$.
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\end{theorem}
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\begin{proof}
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(i) was dealt with in \autoref{roptionalstoppingi}.
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(i) was already done in \autoref{roptionalstoppingi}.
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(ii): Since $(X_n)_n$ is bounded, we get that
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\begin{IEEEeqnarray*}{rCl}
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\end{IEEEeqnarray*}
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Thus, we can apply (ii).
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The statement about martingales follows from
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applying this to $(X_n)_n$ and $(-X_n)_n$,
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which are both supermartingales.
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@ -1,4 +1,4 @@
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\lecture{22}{2023-07-04}{Intro Markov Chains II}
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\lecture{22}{2023-07-04}{Introduction Markov Chains II}
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\begin{goal}
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We want to start with the basics of the theory of Markov chains.
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\end{goal}
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@ -1,5 +1,5 @@
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\lecture{5}{2023-04-21}{}
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\subsection{The laws of large numbers}
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\subsection{The Laws of Large Numbers}
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We want to show laws of large numbers:
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@ -1,3 +1,4 @@
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\lecture{6}{}{}
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\todo{Large parts of lecture 6 are missing}
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\begin{refproof}{lln}
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We want to deduce the SLLN (\autoref{lln}) from \autoref{thm2}.
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@ -1,9 +1,10 @@
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% TODO \begin{goal}
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% TODO We want to drop our assumptions on finite mean or variance
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% TODO and say something about the behaviour of $ \sum_{n \ge 1} X_n$
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% TODO when the $X_n$ are independent.
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% TODO \end{goal}
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\begin{theorem}[Theorem 3, Kolmogorov's three-series theorem] % Theorem 3
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\lecture{7}{}{Kolmogorov's three series theorem}
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\begin{goal}
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We want to drop our assumptions on finite mean or variance
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and say something about the behaviour of $ \sum_{n \ge 1} X_n$
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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{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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|
@ -20,7 +21,7 @@
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\end{enumerate}
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\end{theorem}
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For the proof we'll need a slight generalization of \autoref{thm2}:
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\begin{theorem}[Theorem 4] % Theorem 4
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\begin{theorem} %[Theorem 4]
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\label{thm4}
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Let $\{X_n\}_n$ be independent and \vocab{uniformly bounded}
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(i.e. $\exists M < \infty : \sup_n \sup_\omega |X_n(\omega)| \le M$).
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|
@ -166,14 +167,13 @@ More formally:
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However
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\[
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\sum_{n} X_n \frac{1}{n^{\frac{1}{2} + \epsilon}}
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\]
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\]
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where $\bP[X_n = 1] = \bP[X_n = -1] = \frac{1}{2}$
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converges almost surely for all $\epsilon > 0$.
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And
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\[
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\sum_{n} X_n \frac{1}{n^{\frac{1}{2} - \epsilon}}
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\]
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\]
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does not converge.
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\end{example}
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|
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|
@ -24,7 +24,7 @@ of sequences of random variables.
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is again a $\sigma$-algebra, $\cT$ is indeed a $\sigma$-algebra.
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\item We have
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\[
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\cT = \{A \in \cF ~|~ \forall i ~ \exists B \in \cB(\R)^{\otimes \N} : A = \{\omega | (X_i(\omega), X_{i+1}(\omega), \ldots) \in B\} \}. % TODO?
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\cT = \{A \in \cF ~|~ \forall i ~ \exists B \in \cB(\R)^{\otimes \N} : A = \{\omega | (X_i(\omega), X_{i+1}(\omega), \ldots) \in B\} \}.
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\]
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\end{enumerate}
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\end{remark}
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|
@ -146,5 +146,3 @@ for any $k \in \N$.
|
|||
\]
|
||||
hence $\bP[T] \in \{0,1\}$.
|
||||
\end{refproof}
|
||||
|
||||
|
||||
|
|
|
@ -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}
|
||||
|
||||
|
|
|
@ -1,7 +1,7 @@
|
|||
% This section provides a short recap of things that should be known
|
||||
% from the lecture on stochastics.
|
||||
This section provides a short recap of things that should be known
|
||||
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}
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
|
|
@ -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
|
||||
%\chapter{Appendix}
|
||||
\begin{landscape}
|
||||
\section{Appendix}
|
||||
\input{inputs/a_0_distributions.tex}
|
||||
\end{landscape}
|
||||
|
||||
\cleardoublepage
|
||||
\printvocabindex
|
||||
|
|
13
wtheo.sty
13
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{Ber}
|
||||
\DeclareSimpleMathOperator{Exp}
|
||||
\DeclareSimpleMathOperator{Bin} % binomial distribution
|
||||
\DeclareSimpleMathOperator{Geo} % geometric distribution
|
||||
\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]}}}
|
||||
|
|
Loading…
Reference in a new issue