From 296c7b2f559fcc65776863409e84436cb5367a1a Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Fri, 28 Jul 2023 03:45:37 +0200 Subject: [PATCH] yaref --- inputs/lecture_03.tex | 4 +-- inputs/lecture_04.tex | 12 +++---- inputs/lecture_05.tex | 6 ++-- inputs/lecture_06.tex | 13 +++---- inputs/lecture_07.tex | 34 +++++++++--------- inputs/lecture_08.tex | 2 +- inputs/lecture_09.tex | 14 ++++---- inputs/lecture_10.tex | 24 +++++++------ inputs/lecture_11.tex | 10 +++--- inputs/lecture_12.tex | 10 +++--- inputs/lecture_13.tex | 42 +++++++++++----------- inputs/lecture_14.tex | 7 ++-- inputs/lecture_15.tex | 35 ++++++++++--------- inputs/lecture_16.tex | 10 +++--- inputs/lecture_17.tex | 16 ++++----- inputs/lecture_18.tex | 22 ++++++------ inputs/lecture_19.tex | 24 ++++++------- inputs/lecture_20.tex | 32 ++++++++++++----- inputs/lecture_21.tex | 4 +-- inputs/lecture_23.tex | 75 ++++++++++++++++++++-------------------- inputs/prerequisites.tex | 19 +++++----- jrpie-yaref.sty | 49 ++++++++++++++++++++++++++ wtheo.sty | 1 + 23 files changed, 267 insertions(+), 198 deletions(-) create mode 100644 jrpie-yaref.sty diff --git a/inputs/lecture_03.tex b/inputs/lecture_03.tex index b4436bd..81cfd55 100644 --- a/inputs/lecture_03.tex +++ b/inputs/lecture_03.tex @@ -66,7 +66,7 @@ Define $\cF = \bigcup_{n \in \N} \cF_n$. Then $\cF$ is an algebra. We'll show that if we define $\lambda: \cF \to [0,1]$ with $\lambda(A) = \lambda_n(A)$ for any $n$ where this is well defined, then $\lambda$ is countably additive on $\cF$. -Using \autoref{caratheodory}, $\lambda$ will extend uniquely to a probability measure on $\sigma(\cF)$. +Using \yaref{caratheodory}, $\lambda$ will extend uniquely to a probability measure on $\sigma(\cF)$. We want to prove: \begin{claim} @@ -107,7 +107,7 @@ We want to prove: thus $\cF \subseteq \cB_\infty$. Since $\cB_\infty$ is a $\sigma$-algebra, $\sigma(\cF) \subseteq \cB_\infty$. \end{refproof} -For the proof of \autoref{claim:lambdacountadd}, +For the proof of \yaref{claim:lambdacountadd}, we are going to use the following: \begin{fact} \label{fact:finaddtocountadd} diff --git a/inputs/lecture_04.tex b/inputs/lecture_04.tex index fb244d7..e20c1d0 100644 --- a/inputs/lecture_04.tex +++ b/inputs/lecture_04.tex @@ -1,6 +1,6 @@ \lecture{4}{}{End of proof of Kolmogorov's consistency theorem} -To finish the proof of \autoref{claim:lambdacountadd}, +To finish the proof of \yaref{claim:lambdacountadd}, we need the following: \begin{fact} \label{lec4fact1} @@ -91,7 +91,7 @@ we need the following: so $\{x_k^{(n)}\}_n$ is bounded. \end{itemize} - By \autoref{lec4fact1}, + By \yaref{lec4fact1}, there is an infinite set $S \subseteq \N$, such that $\{x_k^{(n)}\}_{n \in S}$ converges for every $k$. @@ -115,7 +115,7 @@ we need the following: \end{refproof} \begin{refproof}{claim:lambdacountadd} - In order to apply \autoref{fact:finaddtocountadd}, + In order to apply \yaref{fact:finaddtocountadd}, we need the following: \begin{claim} For any sequence $B_n \in \cF$ @@ -163,7 +163,7 @@ we need the following: \[ \bigcap_{k=1}^n L_k^\ast \neq \emptyset. \] - By \autoref{lem:intersectioncompactsets}, + By \yaref{lem:intersectioncompactsets}, it follows that \[ \bigcap_{k \in \N} L_k^\ast \neq \emptyset. @@ -190,7 +190,7 @@ hence \end{IEEEeqnarray*} For the definition of $\lambda$ -as well as the proof of \autoref{claim:lambdacountadd} +as well as the proof of \yaref{claim:lambdacountadd} we have only used that $(\lambda_n)_{n \in \N}$ is a consistent family. -Hence we have in fact shown \autoref{thm:kolmogorovconsistency}. +Hence we have in fact shown \yaref{thm:kolmogorovconsistency}. diff --git a/inputs/lecture_05.tex b/inputs/lecture_05.tex index e927114..9aaeea6 100644 --- a/inputs/lecture_05.tex +++ b/inputs/lecture_05.tex @@ -9,7 +9,7 @@ The RHS is constant, which we can explicitly compute from the distribution of th We fix a probability space $(\Omega, \cF, \bP)$ once and for all. \begin{theorem} - \label{lln} + \label{lln} % TODO - yaref Let $X_1, X_2,\ldots$ be i.i.d.~random variables on $(\R, \cB(\R))$ and $m = \bE[X_i] < \infty$ and $\sigma^{2} = \Var(X_i) = \bE[ (X_i - \bE(X_i))^2] = \bE[X_i^2] - \bE[X_i]^2 < \infty$. @@ -37,7 +37,7 @@ We fix a probability space $(\Omega, \cF, \bP)$ once and for all. \begin{IEEEeqnarray*}{rCl} \bP\left[ \left| \frac{X_1 + \ldots + X_n}{n} - m\right| > \epsilon\right] &=& \bP\left[\left|\frac{S_n}{n}-m\right| > \epsilon\right]\\ - &\overset{\text{Chebyshev}}{\le }& + &\overset{\yaref{thm:chebyshev}}{\le }& \frac{\Var\left( \frac{S_n}{n} \right) }{\epsilon^2} = \frac{1}{n} \frac{\Var(X_1)}{\epsilon^2} \xrightarrow{n \to \infty} 0 @@ -69,7 +69,7 @@ Consider the following: where $X_n$ has distribution $\frac{1}{n^2} \delta_n + \frac{1}{n^2} \delta_{-n} + (1-\frac{2}{n^2}) \delta_0$. We have $\bP[X_n \neq 0] = \frac{2}{n^2}$. - Since this is summable, Borel-Cantelli yields + Since this is summable, \yaref{thm:borelcantelli} yields \[ \bP[X_{n} \neq 0 \text{ for infinitely many $n$}] = 0. \] diff --git a/inputs/lecture_06.tex b/inputs/lecture_06.tex index 414310b..a3584ec 100644 --- a/inputs/lecture_06.tex +++ b/inputs/lecture_06.tex @@ -1,6 +1,6 @@ \lecture{6}{}{Proof of SLLN} \begin{refproof}{lln} - We want to deduce the SLLN (\autoref{lln}) from \autoref{thm2}. + We want to deduce the SLLN (\yaref{lln}) from \yaref{thm2}. W.l.o.g.~let us assume that $\bE[X_i] = 0$ (otherwise define $X'_i \coloneqq X_i - \bE[X_i]$). We will show that $\frac{S_n}{n} \xrightarrow{a.s.} 0$. @@ -8,7 +8,7 @@ Then the $Y_i$ are independent and we have $\bE[Y_i] = 0$ and $\Var(Y_i) = \frac{\sigma^2}{i^2}$. Thus $\sum_{i=1}^\infty \Var(Y_i) < \infty$. - From \autoref{thm2} we obtain that $\sum_{i=1}^\infty Y_i$ converges a.s. + From \yaref{thm2} we obtain that $\sum_{i=1}^\infty Y_i$ converges a.s. \begin{claim} Let $(a_n)$ be a sequence in $\R$ such that $\sum_{n=1}^{\infty} \frac{a_n}{n}$ converges, @@ -45,9 +45,9 @@ The SLLN follows from the claim. \end{refproof} -In order to prove \autoref{thm2}, we need the following: +In order to prove \yaref{thm2}, we need the following: \begin{theorem}[Kolmogorov's inequality] - \label{thm:kolmogorovineq} + \yalabel{Kolmogorov's Inequality}{Kolmogorov}{thm:kolmogorovineq} If $X_1,\ldots, X_n$ are independent with $\bE[X_i] = 0$ and $\Var(X_i) = \sigma_i^2$, then \[ @@ -139,7 +139,7 @@ In order to prove \autoref{thm2}, we need the following: \begin{IEEEeqnarray*}{rCl} &&\max \{|S_{m+1} - S_m|, |S_{m+2} - S_m|, \ldots, |S_{m+l} - S_m|\}\\ &=& \max \{|X_{m+1}|, |X_{m+1} + X_{m+2}|, \ldots, |X_{m+1} + X_{m+2} + \ldots + X_{m+l}|\}\\ - &\overset{\text{\autoref{thm:kolmogorovineq}}}{\le}& + &\overset{\yaref{thm:kolmogorovineq}}{\le}& \frac{1}{\epsilon^2} \sum_{i=m}^{l} \Var(X_i)\\ &\le & \frac{1}{\epsilon^2} \sum_{i=m}^\infty \Var(X_i) \xrightarrow{m \to \infty} 0, @@ -200,6 +200,3 @@ In order to prove \autoref{thm2}, we need the following: Hence $\frac{t}{N_t} \to m$. \end{proof} - - - diff --git a/inputs/lecture_07.tex b/inputs/lecture_07.tex index c63f99a..ed4b62d 100644 --- a/inputs/lecture_07.tex +++ b/inputs/lecture_07.tex @@ -5,7 +5,7 @@ when the $X_n$ are independent. \end{goal} \begin{theorem}[Kolmogorov's three-series theorem] % Theorem 3 - \label{thm:kolmogorovthreeseries} + \yalabel{Kolmogorov's Three-Series Theorem}{3 Series}{thm:kolmogorovthreeseries} \label{thm3} Let $X_n$ be a family of independent random variables. \begin{enumerate}[(a)] @@ -21,7 +21,7 @@ Then all three series above converge for every $C > 0$. \end{enumerate} \end{theorem} -For the proof we'll need a slight generalization of \autoref{thm2}: +For the proof we'll need a slight generalization of \yaref{thm2}: \begin{theorem} %[Theorem 4] \label{thm4} Let $\{X_n\}_n$ be independent and \vocab{uniformly bounded} @@ -31,7 +31,7 @@ For the proof we'll need a slight generalization of \autoref{thm2}: converge. \end{theorem} \begin{refproof}{thm3} - Assume, that we have already proved \autoref{thm4}. + Assume, that we have already proved \yaref{thm4}. We prove part (a) first. Put $Y_n = X_n \cdot \One_{\{|X_n| \le C\}}$. Since the $X_n$ are independent, the $Y_n$ are independent as well. @@ -40,11 +40,11 @@ For the proof we'll need a slight generalization of \autoref{thm2}: $\sum_{n \ge 1} \int_{|X_n| \le C} X_n \dif\bP = \sum_{n \ge 1} \bE[Y_n]$ and $\sum_{n \ge 1} \int_{|X_n| \le C} X_n^2 \dif\bP - \left( \int_{|X_n| \le C} X_n \dif\bP \right)^2 = \sum_{n \ge 1} \Var(Y_n)$ converges. - By \autoref{thm4} it follows that $\sum_{n \ge 1} Y_n < \infty$ + By \yaref{thm4} it follows that $\sum_{n \ge 1} Y_n < \infty$ almost surely. Let $A_n \coloneqq \{\omega : |X_n(\omega)| > C\}$. Since $\sum_{n \ge 1} \bP(A_n) < \infty$ by assumption, - Borel-Cantelli yields $\bP[\text{infinitely many $A_n$ occur}] = 0$. + \yaref{thm:borelcantelli} yields $\bP[\text{infinitely many $A_n$ occur}] = 0$. For the proof of (b), suppose $\sum_{n\ge 1} X_n(\omega) < \infty$ @@ -59,7 +59,7 @@ For the proof we'll need a slight generalization of \autoref{thm2}: \] Then the $Y_n$ are independent and $\sum_{n \ge 1} Y_n(\omega) < \infty$ almost surely and the $Y_n$ are uniformly bounded. - By \autoref{thm4} $\sum_{n \ge 1} \bE[Y_n]$ and $\sum_{n \ge 1} \Var(Y_n)$ + By \yaref{thm4} $\sum_{n \ge 1} \bE[Y_n]$ and $\sum_{n \ge 1} \Var(Y_n)$ converge. Define \[ @@ -70,7 +70,7 @@ For the proof we'll need a slight generalization of \autoref{thm2}: \] Then the $Z_n$ are independent, uniformly bounded and $\sum_{n \ge 1} Z_n(\omega) < \infty$ almost surely. - By \autoref{thm4} we have + By \yaref{thm4} we have $\sum_{n \ge 1} \bE(Z_n) < \infty$ and $\sum_{n \ge 1} \Var(Z_n) < \infty$. @@ -88,8 +88,8 @@ For the proof we'll need a slight generalization of \autoref{thm2}: $\sum_{n \ge 1} \Var(Z_n)$ to conclude that this series converges as well. \end{refproof} -Recall \autoref{thm2}. -We will see, that the converse of \autoref{thm2} is true if the $X_n$ are uniformly bounded. +Recall \yaref{thm2}. +We will see, that the converse of \yaref{thm2} is true if the $X_n$ are uniformly bounded. More formally: \begin{theorem}[Theorem 5] \label{thm5} @@ -99,14 +99,14 @@ More formally: then $\sum_{n \ge 1} \Var(X_n) < \infty$. \end{theorem} \begin{refproof}{thm4} - Assume we have proven \autoref{thm5}. + Assume we have proven \yaref{thm5}. ``$\impliedby$'' Assume $\{X_n\} $ are independent, uniformly bounded and $\sum_{n \ge 1} \bE(X_n) < \infty$ as well as $\sum_{n \ge 1} \Var(X_n) < \infty$. We need to show that $\sum_{n \ge 1} X_n < \infty$ a.s. Let $Y_n \coloneqq X_n - \bE(X_n)$. Then the $Y_n$ are independent, $\bE(Y_n) = 0$ and $\Var(Y_n) = \Var(X_n)$. - By \autoref{thm2} $\sum_{n \ge 1} Y_n < \infty$ a.s. + By \yaref{thm2} $\sum_{n \ge 1} Y_n < \infty$ a.s. Thus $\sum_{n \ge 1} X_n < \infty$ a.s. ``$\implies$'' We assume that $\{X_n\}$ are independent, uniformly bounded @@ -145,23 +145,23 @@ More formally: $\sum_{n \ge 1} \left(Y_n(\omega, \omega') - Z_n(\omega, \omega') \right)= \sum_{n \ge 1} \left(X_n(\omega) - X_n(\omega')\right)$. Thus $\sum_{n \ge 1} \left( Y_n(\omega, \omega') - Z_n(\omega, \omega') \right) < \infty$ a.s.~on $\Omega_0\otimes\Omega_0$. \end{subproof} - By \autoref{thm5}, $\sum_{n} \Var(X_n) = \frac{1}{2}\sum_{n \ge 1} \Var(Y_n - Z_n) < \infty$ a.s. + By \yaref{thm5}, $\sum_{n} \Var(X_n) = \frac{1}{2}\sum_{n \ge 1} \Var(Y_n - Z_n) < \infty$ a.s. Define $U_n \coloneqq X_n - \bE(X_n)$. Then $\bE(U_n) = 0$ and the $U_n$ are independent and uniformly bounded. We have $\sum_{n} \Var(U_n) = \sum_{n} \Var(X_n) < \infty$. - Thus $\sum_{n} U_n$ converges a.s.~by \autoref{thm2}. + Thus $\sum_{n} U_n$ converges a.s.~by \yaref{thm2}. Since by assumption $\sum_{n} X_n < \infty$ a.s., it follows that $\sum_{n} \bE(X_n) < \infty$. \end{refproof} \begin{remark} - In the proof of \autoref{thm4} - ``$\impliedby$'' is just a trivial application of \autoref{thm2} + In the proof of \yaref{thm4} + ``$\impliedby$'' is just a trivial application of \yaref{thm2} and uniform boundedness was not used. The idea of `` $\implies$ '' will lead to coupling. % TODO ? \end{remark} -A proof of \autoref{thm5} can be found in the notes.\notes -\begin{example}[Application of \autoref{thm4}] +A proof of \yaref{thm5} can be found in the notes.\notes +\begin{example}[Application of \yaref{thm4}] The series $\sum_{n} \frac{1}{n^{\frac{1}{2} + \epsilon}}$ does not converge for $\epsilon < \frac{1}{2}$. However diff --git a/inputs/lecture_08.tex b/inputs/lecture_08.tex index a960a7c..d9018d2 100644 --- a/inputs/lecture_08.tex +++ b/inputs/lecture_08.tex @@ -59,7 +59,7 @@ which does not depend on the realisation of the first $k$ random variables for any $k \in \N$. \begin{theorem}[Kolmogorov's 0-1 law] - \label{kolmogorov01} + \yalabel{Kolmogorov's 0-1 Law}{0-1 Law}{kolmogorov01} Let $X_n, n \in \N$ be a sequence of independent random variables and let $\cT$ denote their tail-$\sigma$-algebra. Then $\cT$ is \vocab{$\bP$-trivial}, i.e.~$\bP[A] \in \{0,1\}$ diff --git a/inputs/lecture_09.tex b/inputs/lecture_09.tex index bdcf4ab..1e64137 100644 --- a/inputs/lecture_09.tex +++ b/inputs/lecture_09.tex @@ -1,7 +1,7 @@ \lecture{9}{}{Percolation, Introduction to characteristic functions} \subsubsection{Application: Percolation} -We will now discuss another application of Kolmogorov's $0-1$-law, percolation. +We will now discuss another application of \yaref{kolmogorov01}, percolation. \begin{definition}[\vocab{Percolation}] Consider the graph with nodes $\Z^d$, $d \ge 2$, where edges from the lattice are added with probability $p$. The added edges are called \vocab[Percolation!Edge!open]{open}; @@ -178,7 +178,7 @@ We have \end{remark} \begin{theorem}[Inversion formula] % thm1 - \label{inversionformula} + \yalabel{Inversion Formula}{Inversion Formula}{inversionformula} Let $(\Omega, \cB(\R), \bP)$ be a probability space. Let $F$ be the distribution function of $\bP$ (i.e.~$F(x) = \bP((-\infty, x])$ for all $x \in \R$ ). @@ -193,7 +193,7 @@ We have We will prove this later. \begin{theorem}[Uniqueness theorem] % thm2 - \label{charfuncuniqueness} + \yalabel{Uniqueness Theorem}{Uniqueness}{charfuncuniqueness} Let $\bP$ and $\Q$ be two probability measures on $(\R, \cB(\R))$. Then $\phi_\bP = \phi_\Q \implies \bP = \Q$. @@ -202,20 +202,20 @@ We will prove this later. from $\phi$. \end{theorem} \begin{refproof}{charfuncuniqueness} - Assume that we have already shown \autoref{inversionformula}. + Assume that we have already shown the \yaref{inversionformula}. Suppose that $F$ and $G$ are the distribution functions of $\bP$ and $\Q$. Let $a,b \in \R$ with $a < b$. Assume that $a $ and $b$ are continuity points of both $F$ and $G$. - By \autoref{inversionformula} we have + By the \yaref{inversionformula} we have \begin{IEEEeqnarray*}{rCl} F(b) - F(a) = G(b) - G(a) \label{eq:charfuncuniquefg} \end{IEEEeqnarray*} - Since $F$ and $G$ are monotonic, \autoref{eq:charfuncuniquefg} + Since $F$ and $G$ are monotonic, \yaref{eq:charfuncuniquefg} holds for all $a < b$ outside a countable set. Take $a_n$ outside this countable set, such that $a_n \ssearrow -\infty$. - Then, \autoref{eq:charfuncuniquefg} implies that + Then, \yaref{eq:charfuncuniquefg} implies that $F(b) - F(a_n) = G(b) - G(a_n)$ hence $F(b) = G(b)$. Since $F$ and $G$ are right-continuous, it follows that $F = G$. \end{refproof} diff --git a/inputs/lecture_10.tex b/inputs/lecture_10.tex index 3c6896e..d4a1faf 100644 --- a/inputs/lecture_10.tex +++ b/inputs/lecture_10.tex @@ -14,7 +14,7 @@ where $\mu = \bP X^{-1}$. \begin{refproof}{inversionformula} - We will prove that the limit in the RHS of \autoref{invf} + We will prove that the limit in the RHS of \yaref{invf} exists and is equal to the LHS. Note that the term on the RHS is integrable, as \[ @@ -31,7 +31,7 @@ where $\mu = \bP X^{-1}$. &=& \lim_{T \to \infty} \frac{1}{2 \pi} \int_{\R} \underbrace{\int_{-T}^T \left[ \frac{\cos(t (x-b)) - \cos(t(x-a))}{-\i t}\right] \dif t}_{=0 \text{, as the function is odd}} \bP(\dif x) \\ && + \lim_{T \to \infty} \frac{1}{2\pi} \int_{\R}\int_{-T}^T \frac{\sin(t ( x - b)) - \sin(t(x-a))}{-t} \dif t \bP(\dif x)\\ &=& \lim_{T \to \infty} \frac{1}{\pi} \int_\R \int_{0}^T \frac{\sin(t(x-a)) - \sin(t(x-b))}{t} \dif t \bP(\dif x)\\ - &\overset{\substack{\text{\autoref{fact:sincint},}\\\text{dominated convergence}}}{=}& + &\overset{\substack{\yaref{fact:sincint},\text{dominated convergence}}}{=}& \frac{1}{\pi} \int -\frac{\pi}{2} \One_{x < a} + \frac{\pi}{2} \One_{x > a} - (- \frac{\pi}{2} \One_{x < b} + \frac{\pi}{2} \One_{x > b}) \bP(\dif x)\\ &=& \frac{1}{2} \bP(\{a\} ) + \frac{1}{2} \bP(\{b\}) + \bP((a,b))\\ @@ -103,7 +103,7 @@ where $\mu = \bP X^{-1}$. \bP\left( (a,b] \right) = \int_a^b f(x) \dif x.\label{thm10_3eq1} \] Let $F$ be the distribution function of $\bP$. - It is enough to prove \autoref{thm10_3eq1} + It is enough to prove \yaref{thm10_3eq1} for all continuity points $a $ and $ b$ of $F$. We have \begin{IEEEeqnarray*}{rCl} @@ -112,12 +112,14 @@ where $\mu = \bP X^{-1}$. &=& \frac{1}{2\pi} \int_{\R} \phi(t) \left( \frac{e^{-\i t b} - e^{-\i t a}}{- \i t} \right) \dif t\\ &\overset{\text{dominated convergence}}{=}& \lim_{T \to \infty} \frac{1}{2\pi} \int_{-T}^{T} \phi(t) \left( \frac{e^{-\i t b} - e^{- \i t a}}{- \i t} \right) \dif t \end{IEEEeqnarray*} - By \autoref{inversionformula}, the RHS is equal to $F(b) - F(a) = \bP\left( (a,b] \right)$. + By the \yaref{inversionformula}, + the RHS is equal to $F(b) - F(a) = \bP\left( (a,b] \right)$. \end{refproof} However, Fourier analysis is not only useful for continuous probability density functions: -\begin{theorem}[Bochner's formula for the mass at a point]\label{bochnersformula} % Theorem 4 +\begin{theorem}[Bochner's formula for the mass at a point] + \yalabel{Bochner's Formula for the Mass at a Point}{Bochner}{bochnersformula} % Theorem 4 Let $\bP \in M_1(\lambda)$. Then \[ @@ -174,13 +176,14 @@ However, Fourier analysis is not only useful for continuous probability density &=& \int_{\R} \left| \sum_{l} c_l e^{\i t_l x}\right|^2 \ge 0 \end{IEEEeqnarray*} \end{refproof} -\begin{theorem}[Bochner's theorem]\label{thm:bochner} - The converse to \autoref{thm:lec_10thm5} holds, i.e.~any - $\phi: \R \to \C$ satisfying (a) and (b) of \autoref{thm:lec_10thm5} +\begin{theorem}[Bochner's theorem] + \yalabel{Bochner's Theorem for Positive Definite Functions}{Bochner's Theorem}{thm:bochner}% + The converse to \yaref{thm:lec_10thm5} holds, i.e.~any + $\phi: \R \to \C$ satisfying (a) and (b) of \yaref{thm:lec_10thm5} must be the Fourier transform of a probability measure $\bP$ on $(\R, \cB(\R))$. \end{theorem} -Unfortunately, we won't prove \autoref{thm:bochner} in this lecture. +Unfortunately, we won't prove \yaref{thm:bochner} in this lecture. \begin{definition}[Convergence in distribution / weak convergence] @@ -325,7 +328,8 @@ for all $f \in C_b(\R)$. % \end{itemize} % % \end{proof} -\begin{theorem}[Levy's continuity theorem]\label{levycontinuity} +\begin{theorem}[Levy's continuity theorem] + \yalabel{Levy's Continuity Theorem}{Levy}{levycontinuity} % Theorem 2 $X_n \xrightarrow{\text{d}} X$ iff $\phi_{X_n}(t) \to \phi(t)$ for all $t \in \R$. diff --git a/inputs/lecture_11.tex b/inputs/lecture_11.tex index 06e49dd..670ed07 100644 --- a/inputs/lecture_11.tex +++ b/inputs/lecture_11.tex @@ -52,7 +52,8 @@ In order to make things nicer, we do the following: Then $\bE[\frac{S_n - \bE[S_n]}{\sqrt{\Var(S_n)}}] = 0$ and $\Var(\frac{S_n - \bE[S_n]}{\sqrt{\Var(S_n)}}) = 1$. -\begin{theorem}[Central limit theorem, 1920s, Lindeberg and Levy]\label{clt} +\begin{theorem}[Central limit theorem, 1920s, Lindeberg and Levy]% + \yalabel{Central Limit Theorem}{CLT}{clt} Let $X_1,X_2,\ldots$ be i.i.d.~random variables with $\bE[X_1] = \mu$ and $\Var(X_1) = \sigma^2 \in (0, \infty)$. @@ -81,9 +82,9 @@ There exists a special case of this theorem, which was proved earlier: Let $X_1, X_2,\ldots$ i.i.d.~with $X_1 \sim \Ber(p)$. Then $\bE[X_1] = p$ and $\Var(X_1) = p(1-p )$. Furthermore $\sum_{i=1}^n X_i \sim \Bin(n,p)$, - and the special case follows from \autoref{clt}. + and the special case follows from \yaref{clt}. \end{proof} -\autoref{preclt} is a useful tool for approximating the Binomial distribution with the normal distribution. +\yaref{preclt} is a useful tool for approximating the Binomial distribution with the normal distribution. If $S_n \sim \Bin(n,p)$ and $[a,b] \subseteq \R$, we have \[\bP[a \le S_n \le b] = \bP\left[\frac{a - np}{\sqrt{np(1-p)}} \le \frac{S_n -np}{\sqrt{n p (1-p)}} \le \frac{b - np}{\sqrt{n p (1-p)} }\right] \approx \Phi(b') - \Phi(a').\] @@ -105,7 +106,7 @@ If $S_n \sim \Bin(n,p)$ and $[a,b] \subseteq \R$, we have More formally: Let $X_1,X_2,\ldots$ be i.i.d.~with $\bP[X_1=1] = \bP[X_1=-1] = \frac{1}{2}$ and consider $S_n \coloneqq \sum_{i=1}^n X_i$. - Then \autoref{clt} states, that $S_n \approx \cN(0,n)$. + Then the \yaref{clt} states, that $S_n \approx \cN(0,n)$. \end{example} \begin{example} @@ -129,4 +130,3 @@ If $S_n \sim \Bin(n,p)$ and $[a,b] \subseteq \R$, we have We have $p\cdot (1-p) \le \frac{1}{4}$, thus $n \approx (1.96)^2 \cdot 100^2 \cdot \frac{1}{4} = 9600$ suffices. \end{example} - diff --git a/inputs/lecture_12.tex b/inputs/lecture_12.tex index 350c82b..fc86e9c 100644 --- a/inputs/lecture_12.tex +++ b/inputs/lecture_12.tex @@ -1,12 +1,12 @@ \lecture{12}{2023-05-16}{Proof of the CLT} -We now want to prove \autoref{clt}. +We now want to prove the \yaref{clt}. The plan is to do the following: \begin{enumerate}[1.] \item Identify the characteristic function of a standard normal \item Show that the characteristic functions of the $V_n$ converge pointwise to that of $\cN$. - \item Apply \autoref{levycontinuity} + \item Apply \yaref{levycontinuity} \end{enumerate} First, we need to prove some properties of characteristic functions. @@ -94,7 +94,7 @@ First, we need to prove some properties of characteristic functions. For arbitrary $h \in \R$, we have \begin{IEEEeqnarray*}{rCl} |e^{\i t X} \frac{e^{\i h X}}{h}| &\le & \left| \frac{1}{h} \left( e^{\i h X} - 1 \right)\right|\\ - &\overset{\text{\autoref{charfprop:c1}}}{\le}& \left|\frac{1}{h} \i h X\right| = |X|. + &\overset{\yaref{charfprop:c1}}{\le}& \left|\frac{1}{h} \i h X\right| = |X|. \end{IEEEeqnarray*} Thus the dominated convergence theorem can be applied and we obtain \[ @@ -156,7 +156,7 @@ First, we need to prove some properties of characteristic functions. \end{refproof} -Now, we can finally prove the CLT: +Now, we can finally prove the \yaref{clt}: \begin{refproof}{clt} Let $X_1,X_2,\ldots$ be i.i.d.~random variables with $\bE[X_1] = \mu_1$, $\Var(X_1) = \sigma^2$. @@ -212,7 +212,7 @@ Now, we can finally prove the CLT: \[ \phi_n(t) \xrightarrow{n \to \infty} e^{-\frac{t^2}{2}} = \phi_{\cN(0,1)}(t). \] - Using \autoref{levycontinuity}, we obtain \autoref{clt}. + Using \yaref{levycontinuity}, we obtain the \yaref{clt}. \end{refproof} \begin{remark} diff --git a/inputs/lecture_13.tex b/inputs/lecture_13.tex index df4ef00..e8933f2 100644 --- a/inputs/lecture_13.tex +++ b/inputs/lecture_13.tex @@ -1,5 +1,5 @@ \lecture{13}{2023-05}{} -%The difficult part is to show \autoref{levycontinuity}. +%The difficult part is to show \yaref{levycontinuity}. %This is the last lecture, where we will deal with independent random variables. We have seen, that @@ -12,7 +12,7 @@ if $X_1, X_2,\ldots$ are i.i.d.~with $ \mu = \bE[X_1]$, \end{question} \begin{theorem}[Lindeberg CLT] - \label{lindebergclt} + \yalabel{Lindeberg's CLT}{Lindeberg CLT}{lindebergclt} Assume $X_1, X_2, \ldots,$ are independent (but not necessarily identically distributed) with $\mu_i = \bE[X_i] < \infty$ and $\sigma_i^2 = \Var(X_i) < \infty$. Let $S_n = \sqrt{\sum_{i=1}^{n} \sigma_i^2}$ and assume that @@ -29,7 +29,7 @@ if $X_1, X_2,\ldots$ are i.i.d.~with $ \mu = \bE[X_1]$, \end{theorem} \begin{theorem}[Lyapunov condition] - \label{lyapunovclt} + \yalabel{Lyapunov's CLT}{Lyapunov CLT}{lyapunovclt} Let $X_1, X_2,\ldots$ be independent, $\mu_i = \bE[X_i] < \infty$, $\sigma_i^2 = \Var(X_i) < \infty$ and $S_n \coloneqq \sqrt{\sum_{i=1}^n \sigma_i^2}$. @@ -45,10 +45,10 @@ if $X_1, X_2,\ldots$ are i.i.d.~with $ \mu = \bE[X_1]$, The Lyapunov condition implies the Lindeberg condition. (Exercise). \end{remark} -We will not prove the \autoref{lindebergclt} or \autoref{lyapunovclt} +We will not prove \yaref{lindebergclt} or \yaref{lyapunovclt} in this lecture. However, they are quite important. -We will now sketch the proof of \autoref{levycontinuity}, +We will now sketch the proof of \yaref{levycontinuity}, details can be found in the notes.\notes \begin{definition} Let $(X_n)_n$ be a sequence of random variables. @@ -62,8 +62,8 @@ details can be found in the notes.\notes \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}] +A generalized version of \yaref{levycontinuity} is the following: +\begin{theorem}[A generalized version of \yaref{levycontinuity}] \label{genlevycontinuity} Suppose we have random variables $(X_n)_n$ such that $\bE[e^{\i t X_n}] \xrightarrow{n \to \infty} \phi(t)$ for all $t \in \R$ @@ -77,12 +77,12 @@ A generalized version of \autoref{levycontinuity} is the following: \item $\phi$ is continuous at $0$. \end{enumerate} \end{theorem} -\todo{Proof of \autoref{genlevycontinuity} (Exercise 8.2)} +\todo{Proof of \yaref{genlevycontinuity} (Exercise 8.2)} \begin{example} Let $Z \sim \cN(0,1)$ and $X_n \coloneqq n Z$. We have $\phi_{X_n}(t) = \bE[[e^{\i t X_n}] = e^{-\frac{1}{2} t^2 n^2} \xrightarrow{n \to \infty} \One_{\{t = 0\} }$. $\One_{\{t = 0\}}$ is not continuous at $0$. - By \autoref{genlevycontinuity}, $X_n$ can not converge to a real-valued + By \yaref{genlevycontinuity}, $X_n$ can not converge to a real-valued random variable. Exercise: $X_n \xrightarrow{(d)} \overline{X}$, @@ -102,12 +102,12 @@ A generalized version of \autoref{levycontinuity} is the following: \frac{1}{n} \bE[ (X_1+ \ldots + X_n)^2]\\ &=& \sigma^2 \end{IEEEeqnarray*} - For $a > 0$, by Chebyshev's inequality, % TODO + For $a > 0$, by \yaref{thm:chebyshev}, we have \[ \bP\left[ \left| \frac{S_n}{\sqrt{n}} \right| > a \right] \leq \frac{\sigma^2}{a^2} \xrightarrow{a \to \infty} 0. \] - verifying \autoref{genlevycontinuity}. + verifying \yaref{genlevycontinuity}. \end{example} \begin{example} @@ -133,9 +133,9 @@ A generalized version of \autoref{levycontinuity} is the following: Exercise: $\phi_{\frac{S_n}{n}}(t) = e^{-|t|} = \phi_{C_1}(t)$, thus $\frac{S_n}{n} \sim C$. \end{example} -We will prove \autoref{levycontinuity} assuming -\autoref{lec10_thm1}. -\autoref{lec10_thm1} will be shown in the notes.\notes +We will prove \yaref{levycontinuity} assuming +\yaref{lec10_thm1}. +\yaref{lec10_thm1} will be shown in the notes.\notes We will need the following: \begin{lemma} \label{lec13_lem1} @@ -217,7 +217,7 @@ for all $t \in \R$. Apply dominated convergence. \end{subproof} So to prove $\mu_n\left( (-A,A) \right) \ge 1 - 2 \epsilon$, - apply \autoref{s7e1}. + apply \yaref{s7e1}. It suffices to show that \[ \frac{A}{2} \left| \int_{-\frac{2}{A}}^{\frac{2}{A}} \phi_n(t) dt\right| - 1 \ge 1 - 2\epsilon @@ -226,11 +226,11 @@ for all $t \in \R$. \[ 1 - \frac{A}{4} \left|\int_{-\frac{2}{A}}^{\frac{2}{A}} \phi_n(t) dt \right| \le \epsilon, \] - which follows from \autoref{levyproofc1eqn2}. + which follows from \yaref{levyproofc1eqn2}. \end{refproof} % Step 2 -By \autoref{lec13_lem1} +By \yaref{lec13_lem1} there exists a right continuous, non-decreasing $F $ and a subsequence $(F_{n_k})_k$ of $(F_n)_n$ where $F_n$ is the probability distribution function of $\mu_n$, @@ -251,7 +251,7 @@ such that $F_{n_k}(x) \to F(x)$ for all $x$ where $F$ is continuous. \mu_{n_k}\left( (- \infty, x] \right) = F_{n_k}(x) \to F(x). \] Again, given $\epsilon > 0$, there exists $A > 0$, such that - $\mu_{n_k}\left( (-A,A) \right) > 1 - 2 \epsilon$ (\autoref{levyproofc1}). + $\mu_{n_k}\left( (-A,A) \right) > 1 - 2 \epsilon$ (\yaref{levyproofc1}). Hence $F(x) \ge 1 - 2 \epsilon$ for $x > A $ and $F(x) \le 2\epsilon$ for $x < -A$. @@ -262,13 +262,13 @@ Since $F$ is a probability distribution function, there exists a probability measure $\nu$ on $\R$ such that $F$ is the distribution function of $\nu$. Since $F_{n_k}(x) \to F_n(x)$ at all continuity points $x$ of $F$, -by \autoref{lec10_thm1} we obtain that +by \yaref{lec10_thm1} we obtain that $\mu_{n_k} \overset{k \to \infty}{\implies} \nu$. Hence $\phi_{\mu_{n_k}}(t) \to \phi_\nu(t)$, by the other direction of that theorem. But by assumption, $\phi_{\mu_{n_k}}(\cdot ) \to \phi_n(\cdot )$ so $\phi_{\mu}(\cdot) = \phi_{\nu}(\cdot )$. -By \autoref{charfuncuniqueness}, we get $\mu = \nu$. +By the \yaref{charfuncuniqueness}, we get $\mu = \nu$. We have shown, that $\mu_{n_k} \implies \mu$ along a subsequence. We still need to show that $\mu_n \implies \mu$. @@ -281,7 +281,7 @@ We still need to show that $\mu_n \implies \mu$. % \notes % \end{subproof} Assume that $\mu_n$ does not converge to $\mu$. -By \autoref{lec10_thm1}, pick a continuity point $x_0$ of $F$, +By \yaref{lec10_thm1}, pick a continuity point $x_0$ of $F$, such that $F_n(x_0) \not\to F(x_0)$. Pick $\delta > 0$ and a subsequence $F_{n_1}(x_0), F_{n_2}(x_0), \ldots$ which are all outside $(F(x_0) - \delta, F(x_0) + \delta)$. diff --git a/inputs/lecture_14.tex b/inputs/lecture_14.tex index 91095a3..37eb754 100644 --- a/inputs/lecture_14.tex +++ b/inputs/lecture_14.tex @@ -89,7 +89,7 @@ We now want to generalize this to arbitrary random variables. \subsection{Existence of Conditional Probability} -We will give two different proves of \autoref{conditionalexpectation}. +We will give two different proves of \yaref{conditionalexpectation}. The first one will use orthogonal projections. The second will use the Radon-Nikodym theorem. We'll first do the easy proof, derive some properties @@ -139,7 +139,7 @@ and then do the harder proof. $K$ is closed, since a pointwise limit of $\cG$-measurable functions is $\cG$ measurable (if it exists). - By \autoref{orthproj}, + By \yaref{orthproj}, there exists $z \in K$ such that \[\bE[(X - Z)^2] = \inf \{ \bE[(X- W)^2] ~|~ W \in L^2(\cG)\}\] and @@ -168,5 +168,6 @@ and then do the harder proof. Define $Z(\omega) \coloneqq \limsup_{n \to \infty} Z_n(\omega)$. Then $Z$ is $\cG$-measurable and since $Z_n \uparrow Z$, - by MCT, $\bE(Z \One_G) = \bE(X \One_G)$ for all $G \in \cG$. + by the \yaref{cmct}, + $\bE(Z \One_G) = \bE(X \One_G)$ for all $G \in \cG$. \end{refproof} diff --git a/inputs/lecture_15.tex b/inputs/lecture_15.tex index 8e5385a..3bb4c76 100644 --- a/inputs/lecture_15.tex +++ b/inputs/lecture_15.tex @@ -5,7 +5,7 @@ We want to derive some properties of conditional expectation. \begin{theorem}[Law of total expectation] \label{ceprop1} - \label{totalexpectation} + \yalabel{Law of Total Expectation}{Total Expectation}{totalexpectation} \[ \bE[\bE[X | \cG ]] = \bE[X]. \] @@ -26,7 +26,7 @@ We want to derive some properties of conditional expectation. \[ \int_A X \dif \bP \ge \frac{1}{n}\bP(A) + \int_A Y \dif \bP, \] - contradicting property (b) from \autoref{conditionalexpectation}. + contradicting property (b) from \yaref{conditionalexpectation}. \end{proof} \begin{example} @@ -37,7 +37,7 @@ We want to derive some properties of conditional expectation. \begin{theorem}[Linearity] \label{ceprop3} - \label{celinearity} + \yalabel{Linearity of Conditional Expectation}{Linearity}{celinearity} For all $a,b \in \R$ we have \[ @@ -50,7 +50,7 @@ We want to derive some properties of conditional expectation. \begin{theorem}[Positivity] \label{ceprop4} - \label{cpositivity} + \yalabel{Positivity of Conditional Expectation}{Positivity}{cpositivity} If $X \ge 0$, then $\bE[X | \cG] \ge 0$ a.s. \end{theorem} \begin{proof} @@ -65,7 +65,7 @@ We want to derive some properties of conditional expectation. \end{proof} \begin{theorem}[Conditional monotone convergence theorem] \label{ceprop5} - \label{mcmt} + \yalabel{Conditional Monotone Converence Theorem}{MCT}{cmct} Let $X_n,X \in L^1(\Omega, \cF, \bP)$. Suppose $X_n \ge 0$ with $X_n \uparrow X$. Then $\bE[X_n|\cG] \uparrow \bE[X|\cG]$. @@ -73,7 +73,7 @@ We want to derive some properties of conditional expectation. \begin{proof} Let $Z_n$ be a version of $\bE[X_n | Y]$. Since $X_n \ge 0$ and $X_n \uparrow$, - by \autoref{cpositivity}, + by the \yaref{cpositivity}, we have \[ \bE[X_n | \cG] \overset{\text{a.s.}}{\ge } 0 @@ -100,7 +100,7 @@ We want to derive some properties of conditional expectation. \begin{theorem}[Conditional Fatou] \label{ceprop6} - \label{cfatou} + \yalabel{Conditional Fatou's Lemma}{Fatou}{cfatou} Let $X_n \in L^1(\Omega, \cF, \bP)$, $X_n \ge 0$. Then \[ @@ -112,7 +112,7 @@ We want to derive some properties of conditional expectation. \end{proof} \begin{theorem}[Conditional dominated convergence theorem] \label{ceprop7} - \label{cdct} + \yalabel{Conditional Dominated Convergence Theorem}{DCT}{cdct} Let $X_n,Y \in L^1(\Omega, \cF, \bP)$. Suppose that $\sup_n |X_n(\omega)| < Y(\omega)$ a.e.~ and that $X_n$ converges to a pointwise limit $X$. @@ -124,7 +124,7 @@ We want to derive some properties of conditional expectation. Recall \begin{fact}[Jensen's inequality] - \label{jensen} + \yalabel{Jensen's Inequality}{Jensen}{jensen} If $c : \R \to \R$ is convex and $\bE[|c \circ X|] < \infty$, then $\bE[c \circ X] \overset{\text{a.s.}}{\ge} c(\bE[X])$. \end{fact} @@ -132,7 +132,7 @@ Recall For conditional expectation, we have \begin{theorem}[Conditional Jensen's inequality] \label{ceprop8} - \label{cjensen} + \yalabel{Jensen's Inequality}{Jensen}{cjensen} Let $X \in L^1(\Omega, \cF, \bP)$. If $c : \R \to \R$ is convex and $\bE[|c \circ X|] < \infty$, then $\bE[c \circ X | \cG] \ge c(\bE[X | \cG])$ a.s. @@ -147,7 +147,7 @@ For conditional expectation, we have \] \end{fact} \begin{refproof}{cjensen} - By \autoref{convapprox}, $c(x) \ge a_n X + b_n$ + By \yaref{convapprox}, $c(x) \ge a_n X + b_n$ for all $n$. Hence \[ @@ -159,12 +159,13 @@ For conditional expectation, we have we conclude that a.s~this happens simultaneously for all $n$. Hence \[ - \bE[c(X) | \cG] \ge \sup_n (a_n \bE[X | \cG] + b_n) \overset{\text{\autoref{convapprox}}}{=} c(\bE(X | \cG)). + \bE[c(X) | \cG] \ge \sup_n (a_n \bE[X | \cG] + b_n) \overset{\yaref{convapprox}}{=} c(\bE(X | \cG)). \] \end{refproof} Recall \begin{fact}[Hölder's inequality] + \yalabel{Hölder's Inequality}{Hölder}{thm:hoelder} Let $p,q \ge 1$ such that $\frac{1}{p} + \frac{1}{q} = 1$. Suppose $X \in L^p(\bP)$ and $Y \in L^q(\bP)$. Then @@ -175,7 +176,7 @@ Recall \begin{theorem}[Conditional Hölder's inequality] \label{ceprop9} - \label{choelder} + \yalabel{Hölder's Inequality}{Hölder}{choelder} Let $p,q \ge 1$ such that $\frac{1}{p} + \frac{1}{q} = 1$. Suppose $X \in L^p(\bP)$ and $Y \in L^q(\bP)$. Then @@ -203,7 +204,7 @@ Recall \begin{theorem}[Tower property] \label{ceprop10} - \label{cetower} + \yalabel{Tower Property}{Tower}{cetower} Suppose $\cF \supset \cG \supset \cH$ are sub-$\sigma$-algebras. Then \[ @@ -245,7 +246,7 @@ Assume $Y = \One_B$, then $Y$ simple, then take the limit (using that $Y$ is bou \begin{theorem}[Role of independence] \label{ceprop12} - \label{ceroleofindependence} + \yalabel{Role of Independence}{Independence}{ceroleofindependence} Let $X$ be a random variable, and let $\cG, \cH$ be $\sigma$-algebras. @@ -272,10 +273,10 @@ Assume $Y = \One_B$, then $Y$ simple, then take the limit (using that $Y$ is bou For $\bE[S_{n+1} | \cF_n]$ we obtain \begin{IEEEeqnarray*}{rCl} - \bE[S_{n+1} | \cF_n] &\overset{\text{\autoref{celinearity}}}{=}& + \bE[S_{n+1} | \cF_n] &\overset{\yaref{celinearity}}{=}& \bE[S_n | \cF_n] + \bE[X_{n+1} | \cF_n]\\ &\overset{\text{a.s.}}{=}& S_n + \bE[X_{n+1} | \cF_n]\\ - &\overset{\text{\autoref{ceprop12}}}{=}& S_{n} + \bE[X_n]\\ + &\overset{\yaref{ceroleofindependence}}{=}& S_{n} + \bE[X_n]\\ &=& S_n. \end{IEEEeqnarray*} \end{example} diff --git a/inputs/lecture_16.tex b/inputs/lecture_16.tex index f705564..19cc478 100644 --- a/inputs/lecture_16.tex +++ b/inputs/lecture_16.tex @@ -48,7 +48,7 @@ Note that in this setting, if $\mu(A) = 0$ it follows that $\nu(A) = 0$. The Radon Nikodym theorem is the converse of that: \begin{theorem}[Radon-Nikodym] - \label{radonnikodym} + \yalabel{Radon-Nikodym Theorem}{Radon-Nikodym}{radonnikodym} Let $\mu$ and $\nu$ be two $\sigma$-finite measures on $(\Omega, \cF)$. @@ -85,14 +85,14 @@ The Radon Nikodym theorem is the converse of that: \end{definition} -With \autoref{radonnikodym} we get a very short proof of the existence +With the \yaref{radonnikodym} we get a very short proof of the existence of conditional expectation: -\begin{proof}[Second proof of \autoref{conditionalexpectation}] +\begin{proof}[Second proof of \yaref{conditionalexpectation}] Let $(\Omega, \cF, \bP)$ as always, $X \in L^1(\bP)$ and $\cG \subseteq \cF$. It suffices to consider the case of $X \ge 0$. For all $G \in \cG$, define $\nu(G) \coloneqq \int_G X \dif \bP$. Obviously, $\nu \ll \bP$ on $\cG$. - Then apply \autoref{radonnikodym}. + Then apply the \yaref{radonnikodym}. \end{proof} @@ -212,7 +212,7 @@ Typically $\cF_n = \sigma(X_1, \ldots, X_n)$ for a sequence of random variables. Likewise, if $f$ is concave, then $((f(X_n))_n$ is a supermartingale. \end{corollary} \begin{proof} - Apply \autoref{cjensen}. + Apply \yaref{cjensen}. \end{proof} \begin{corollary} diff --git a/inputs/lecture_17.tex b/inputs/lecture_17.tex index c50cc74..a829211 100644 --- a/inputs/lecture_17.tex +++ b/inputs/lecture_17.tex @@ -130,9 +130,9 @@ exists pointwise. \end{lemma} \begin{proof} Since $C_n \ge 0$, - by \autoref{lem:gambling-strategy} we have that $Y_n$ is a supermartingale. + by \yaref{lem:gambling-strategy} we have that $Y_n$ is a supermartingale. Hence $\bE[Y_N] \le \bE[Y_1] = 0$. - From \autoref{lec17l2} it follows that + From \yaref{lec17l2} it follows that \[ (b-a) \bE[U_N([a,b])] \le \bE[Y_n] + \bE[(X_N-a)^-] \le \bE[(X_N-a)^-]. \] @@ -146,7 +146,7 @@ exists pointwise. In particular, $\bP[U_\infty = \infty] = 0$. \end{corollary} \begin{proof} - By \autoref{lec17l3} + By \yaref{lec17l3} we have that \[(b-a) \bE[U_N([a,b])] \le \bE[ | X_N| ] + |a| \le \sup_n \bE[|X_n|] + |a|.\] Since $U_N(\cdot) \ge 0$ and $U_N(\cdot ) \uparrow U_\infty(\cdot )$, @@ -160,8 +160,8 @@ Let us now consider the case that our process $(X_n)_{n \ge 1}$ is a supermartin bounded in $L^1(\bP)$. \begin{theorem}[Doob's martingale convergence theorem] - \label{doobmartingaleconvergence} - \label{doob} + \yalabel{Doob's Martingale Convergence Theorem}{Doob}{doobmartingaleconvergence} + \yalabel{Doob's Martingale Convergence Theorem}{Doob}{doob} Any supermartingale bounded in $L^1$ converges almost surely to a random variable, which is almost surely finite. In particular, any non-negative supermartingale converges a.s.~to a finite random variable. @@ -179,8 +179,8 @@ We have \end{IEEEeqnarray*} We have $\Lambda_{a,b} \subseteq \{\omega : U_{\infty}([a,b])(\omega) = \infty\}$ -by \autoref{lec17l1}. -By \autoref{lec17l3} we have $\bP(\Lambda_{a,b}) = 0$, +by \yaref{lec17l1}. +By \yaref{lec17l3} we have $\bP(\Lambda_{a,b}) = 0$, hence $\bP(\Lambda) = 0$. Thus there exists a random variable $X_\infty$ such that $X_n \xrightarrow{a.s.} X_\infty$. @@ -192,7 +192,7 @@ Thus there exists a random variable $X_\infty$ such that $X_n \xrightarrow{a.s.} We have. \begin{IEEEeqnarray*}{rCl} \bE[|X_\infty|] &=& \bE[\liminf_{n \to \infty} |X_n|]\\ - &\overset{\text{Fatou}}{\le }& \liminf_n \bE[|X_n|]\\ + &\overset{\yaref{cfatou}}{\le }& \liminf_n \bE[|X_n|]\\ &\le & \sup_n \bE[|X_n|]\\ &<& \infty. \end{IEEEeqnarray*} diff --git a/inputs/lecture_18.tex b/inputs/lecture_18.tex index 4df42f5..e8a4489 100644 --- a/inputs/lecture_18.tex +++ b/inputs/lecture_18.tex @@ -37,7 +37,7 @@ Hence the same holds for submartingales, i.e. &=& X_n. \end{IEEEeqnarray*} - By \autoref{doobmartingaleconvergence}, + By \yaref{doobmartingaleconvergence}, there exists an a.s.~limit $X_\infty$. By the SLLN, we have almost surely \[ @@ -116,7 +116,7 @@ consider $L^2$. \[ \bE[X_n^2] = \bE[X_0^2] + \sum_{j=1}^{n} \bE[Y_j^2] \] - by \autoref{martingaleincrementsorthogonal}. + by \yaref{martingaleincrementsorthogonal}. In particular, \[ \sup_n \bE[X_n^2] < \infty \iff \sum_{j=1}^{\infty} \bE[Y_j^2] < \infty. @@ -124,7 +124,7 @@ consider $L^2$. Since $(X_n)_n$ is bounded in $L^2$, there exists $X_\infty$ such that $X_n \xrightarrow{\text{a.s.}} X_\infty$ - by \autoref{doob}. + by \yaref{doob}. It remains to show $X_n \xrightarrow{L^2} X_\infty$. For any $r \in \N$, consider @@ -143,7 +143,7 @@ consider $L^2$. Now let $p \ge 1$ be not necessarily $2$. First, we need a very important inequality: \begin{theorem}[Doob's $L^p$ inequality] - \label{dooblp} + \yalabel{Doob's Martingale Inequalities}{Doob}{dooblp} Suppose that $(X_n)_n$ is a martingale or a non-negative submartingale. Let $X_n^\ast \coloneqq \max \{|X_1|, |X_2|, \ldots, |X_n|\}$ @@ -158,7 +158,7 @@ First, we need a very important inequality: \end{enumerate} \end{theorem} -In order to prove \autoref{dooblp}, we first need +In order to prove \yaref{dooblp}, we first need \begin{lemma} \label{dooplplemma} Let $p > 1$ and $X,Y$ non-negative random variables @@ -182,7 +182,7 @@ In order to prove \autoref{dooblp}, we first need &=& \int Y(\omega)^p \dif \bP(\omega)\\ &=&\int_{\Omega} \left( \int_0^{Y(\omega)} p \ell^{p-1} \dif \ell \right) \dif \bP(\omega)\\ - &\overset{\text{Fubini}}{=}& + &\overset{\yaref{thm:fubini}}{=}& \int_0^\infty p \ell^{p-1}\underbrace{\int_\Omega \One_{Y \ge \ell}\dif \bP}_% {\bP[Y \ge \ell]} \dif\ell. \label{l18star} \end{IEEEeqnarray} @@ -192,7 +192,7 @@ In order to prove \autoref{dooblp}, we first need \eqref{l18star} &\le& \int_0^\infty p \ell^{p-2} \int_{\{Y(\omega) \ge \ell\}} X(\omega) \bP(\dif \omega)\dif \ell\\ - &\overset{\text{Fubini}}{=}& + &\overset{\yaref{thm:fubini}}{=}& \int_\Omega X(\omega) \int_{0}^{Y(\omega)} p \ell^{p-2} \dif \ell\bP(\dif \omega)\\ &=& \frac{p}{p-1} \int_{\omega} X(\omega) Y (\omega)^{p-1} \bP(\dif \omega)\\ &\overset{\text{Hölder}}{\le}& \frac{p}{p-1} \|X\|_{L^p} \|Y\|_{p}^{p-1}, @@ -212,11 +212,11 @@ In order to prove \autoref{dooblp}, we first need \] Then \begin{equation} - \bP[E_j] \overset{\text{Markov}}{\le } \frac{1}{\ell} \int_{E_j} |X_j| \dif \bP + \bP[E_j] \overset{\yaref{thm:markov}}{\le } \frac{1}{\ell} \int_{E_j} |X_j| \dif \bP \label{lec18eq2star} \end{equation} We have that $(|X_n|)_n$ is a submartingale, - by \autoref{cor:convexmartingale} + by \yaref{cor:convexmartingale} in the case of $X_n$ being a martingale and trivially if $X_n$ is non-negative. Hence @@ -225,7 +225,7 @@ In order to prove \autoref{dooblp}, we first need &=& \One_{E_j} \bE[(|X_n| - |X_{j}|)|\cF_j]\\ &\overset{\text{a.s.}}{\ge }& 0. \end{IEEEeqnarray*} - By the law of total expectation, \autoref{totalexpectation}, + By the \yaref{totalexpectation}, it follows that \begin{equation} \bE[\One_{E_j} (|X_n| - |X_j|)] \ge 0. \label{lec18eq3star} @@ -241,6 +241,6 @@ In order to prove \autoref{dooblp}, we first need This proves the first part. For the second part, we apply the first part and - \autoref{dooplplemma} (choose $Y \coloneqq X_n^\ast$). + \yaref{dooplplemma} (choose $Y \coloneqq X_n^\ast$). \end{refproof} \todo{Branching process} diff --git a/inputs/lecture_19.tex b/inputs/lecture_19.tex index 6df66a0..9b6b8bc 100644 --- a/inputs/lecture_19.tex +++ b/inputs/lecture_19.tex @@ -57,10 +57,10 @@ However, some subsets can be easily described, e.g. \sup_n \underbrace{\bP[|X_n| > K]^{\frac{1}{q}}}_% {\le K^{-\frac{1}{q}} \bE[|X_n|]^{\frac{1}{q}}}\\ \end{IEEEeqnarray*} - where we have applied Markov's inequality. % TODO REF + where we have applied \yaref{thm:markov}. Since $\sup_n \bE[|X_n|^{1+\delta}] < \infty$, - we have that $\sup_n \bE[|X_n|] < \infty$ by Jensen (\autoref{jensen}). + we have that $\sup_n \bE[|X_n|] < \infty$ by \yaref{jensen}. Hence for $K$ large enough relevant term is less than $\epsilon$. \end{proof} @@ -93,7 +93,7 @@ However, some subsets can be easily described, e.g. but $\int_{F_n} |X| \dif \bP \ge \epsilon$. Since $\sum_{n} \bP(F_n) < \infty$, - by \autoref{borelcantelli}, + by \yaref{thm:borelcantelli}, \[\bP[\underbrace{\limsup_n F_n}_{\text{\reflectbox{$\coloneqq$}}F}] = 0.\] We have \begin{IEEEeqnarray*}{rCl} @@ -108,7 +108,7 @@ However, some subsets can be easily described, e.g. This yields a contradiction since $\bP(F) = 0$. \item We want to apply part (a) to $F = \{ |X| > k\}$. - By Markov, $\bP(F) \le \frac{1}{k} \bE[|X|]$. + By \yaref{thm:markov}, $\bP(F) \le \frac{1}{k} \bE[|X|]$. Since $\bE[|X|] < \infty$, we can choose $k$ large enough to get $\bP(F) \le \delta$. \end{enumerate} @@ -120,7 +120,7 @@ However, some subsets can be easily described, e.g. \[ \bE[|X_n| \One_{|X_n| > k}] \le \bE[|Y| \One_{|Y| > k}] < \epsilon \] - for $k$ large enough by \autoref{lec19f4} (b). + for $k$ large enough by \yaref{lec19f4} (b). \end{refproof} \begin{fact}\label{lec19f5} @@ -135,9 +135,9 @@ However, some subsets can be easily described, e.g. \label{lec19eqstar} \end{equation} Let $Y = \bE[X | \cG]$ for some sub-$\sigma$-algebra $\cG$. - Then, by \autoref{cjensen}, $|Y| \le \bE[ |X| | \cG]$. + Then, by \yaref{cjensen}, $|Y| \le \bE[ |X| | \cG]$. Hence $\bE[|Y|] \le \bE[|X|]$. - By Markov's inequality, + By \yaref{thm:markov}, it follows that $\bP[|Y| > k] < \delta$ for $k > \frac{\bE[|X|]}{\delta}$. Note that $\{|Y| > k\} \in \cG$. @@ -179,19 +179,19 @@ However, some subsets can be easily described, e.g. + \int |\phi(X_n) - \phi(X)| \dif \bP\\ \end{IEEEeqnarray*} We have $\int_{|X_n| > k} \underbrace{|X_n - \phi(X_n)|}_{\le |X_n| + | \phi(X_n)| \le 2 |X_n|} \dif \bP\le \epsilon$ by uniform integrability and - \autoref{lec19f4} part (b). + \yaref{lec19f4} part (b). Similarly $\int_{|X| > k} |X - \phi(X)| \dif \bP < \epsilon$. Since $\phi$ is Lipschitz, $ X_n \xrightarrow{\bP} X \implies \phi(X_n) \xrightarrow{\bP} \phi(X)$. - By the bounded convergence theorem, \autoref{thm:boundedconvergence}, + By the \yaref{thm:boundedconvergence} $|\phi(X_n)| \le k \implies \int | \phi(X_n) - \phi(X)| \dif \bP \to 0$. (1) $\implies$ (2) $X_n \xrightarrow{L^1} X \implies X_n \xrightarrow{\bP} X$ - by Markov's inequality (see \autoref{claim:convimpll1p}). + by \yaref{thm:markov} (see \yaref{claim:convimpll1p}). Fix $\epsilon > 0$. We have @@ -202,8 +202,8 @@ However, some subsets can be easily described, e.g. \end{IEEEeqnarray*} for all $\delta > 0$ and suitable $k$. - Hence $\bP[|X_n| > k] < \delta$ by Markov's inequality. - Then by \autoref{lec19f4} part (a) it follows that + Hence $\bP[|X_n| > k] < \delta$ by \yaref{thm:markov}. + Then by \yaref{lec19f4} part (a) it follows that \[ \int_{|X_n| > k} |X_n| \dif \bP \le \underbrace{\int |X - X_n| \dif \bP}_{< \epsilon} + \int_{|X_n| > k} |X| \dif \bP \le 2 \epsilon. \] diff --git a/inputs/lecture_20.tex b/inputs/lecture_20.tex index 63c197f..42a78ac 100644 --- a/inputs/lecture_20.tex +++ b/inputs/lecture_20.tex @@ -1,16 +1,16 @@ \lecture{20}{2023-06-27}{} \begin{refproof}{ceismartingale} - By the tower property (\autoref{cetower}) + By the \yaref{cetower} it is clear that $(\bE[X | \cF_n])_n$ is a martingale. First step: Assume that $X$ is bounded. - Then, by \autoref{cjensen}, $|X_n| \le \bE[|X| | \cF_n]$, + Then, by \yaref{cjensen}, $|X_n| \le \bE[|X| | \cF_n]$, hence $\sup_{\substack{n \in \N \\ \omega \in \Omega}} | X_n(\omega)| < \infty$. Thus $(X_n)_n$ is a martingale in $L^{\infty} \subseteq L^2$. By the convergence theorem for martingales in $L^2$ - (\autoref{martingaleconvergencel2}) + (\yaref{martingaleconvergencel2}) there exists a random variable $Y$, such that $X_n \xrightarrow{L^2} Y$. @@ -74,10 +74,10 @@ Thus $X_n \xrightarrow{L^p} X$. \end{refproof} -For the proof of \autoref{martingaleisce}, +For the proof of \yaref{martingaleisce}, we need the following theorem, which we won't prove here: \begin{theorem}[Banach Alaoglu] - \label{banachalaoglu} + \yalabel{Banach Alaoglu}{Banach Alaoglu}{banachalaoglu} Let $X$ be a normed vector space and $X^\ast$ its continuous dual. Then the closed unit ball in $X^\ast$ is compact @@ -96,7 +96,7 @@ we need the following theorem, which we won't prove here: \end{fact} \begin{refproof}{martingaleisce} - Since $(X_n)_n$ is bounded in $L^p$, by \autoref{banachalaoglu}, + Since $(X_n)_n$ is bounded in $L^p$, by \yaref{banachalaoglu}, there exists $X \in L^p$ and a subsequence $(X_{n_k})_k$ such that for all $Y \in L^q$ ($\frac{1}{p} + \frac{1}{q} = 1$ ) \[ @@ -115,7 +115,7 @@ we need the following theorem, which we won't prove here: &\overset{\text{for }n_k \ge m}{=}& \bE[X_m \One_A]. \end{IEEEeqnarray*} Hence $X_n = \bE[X | \cF_m]$ by the uniqueness of conditional expectation - and by \autoref{ceismartingale}, + and by \yaref{ceismartingale}, we get the convergence. \end{refproof} @@ -298,7 +298,7 @@ we need the following theorem, which we won't prove here: \end{example} \begin{theorem}[Optional Stopping] - \label{optionalstopping} + \yalabel{Optional Stopping Theorem}{Optional Stopping}{optionalstopping} Let $(X_n)_n$ be a supermartingale and let $T$ be a stopping time taking values in $\N$. @@ -320,7 +320,7 @@ we need the following theorem, which we won't prove here: $\bE[X_T] = \bE[X_0]$. \end{theorem} \begin{proof} - (i) was already done in \autoref{roptionalstoppingi}. + (i) was already done in \yaref{roptionalstoppingi}. (ii): Since $(X_n)_n$ is bounded, we get that \begin{IEEEeqnarray*}{rCl} @@ -345,3 +345,17 @@ we need the following theorem, which we won't prove here: applying this to $(X_n)_n$ and $(-X_n)_n$, which are both supermartingales. \end{proof} +\begin{remark}+ + Let $(X_n)_n$ be a supermartingale and $T$ a stopping time. + If $(X_n)_n$ itself is not bounded, + but $T$ ensures boundedness, + i.e. $T < \infty$ a.s.~and $(X_{T \wedge n})_n$ + is uniformly bounded, + the \yaref{optionalstopping} can still be applied, as + \[ + \bE[X_T] = \bE[X_{T \wedge T}] + \overset{\yaref{optionalstopping}}{\le} \bE[X_{T \wedge 0}] + = \bE[X_0]. + \] +\end{remark} + diff --git a/inputs/lecture_21.tex b/inputs/lecture_21.tex index faf1325..01c5177 100644 --- a/inputs/lecture_21.tex +++ b/inputs/lecture_21.tex @@ -6,7 +6,7 @@ This is the last lecture relevant for the exam. \begin{goal} We want to see an application of the - optional stopping theorem \ref{optionalstopping}. + \ref{optionalstopping}. \end{goal} \begin{notation} @@ -110,7 +110,7 @@ is the unique solution to this problem. % We have $\sigma(\One_{X_{n+1} \in B}) \subseteq \sigma(X_{n}, \xi_{n+1})$. % $\sigma(X_1,\ldots,X_{n-1})$ % is independent of $\sigma( \sigma(\One_{X_{n+1} \in B}), X_n)$. - % Hence the claim follows from \autoref{ceroleofindependence}. + % Hence the claim follows from \yaref{ceroleofindependence}. \end{subproof} \end{example} diff --git a/inputs/lecture_23.tex b/inputs/lecture_23.tex index 3d52b1e..ee83d87 100644 --- a/inputs/lecture_23.tex +++ b/inputs/lecture_23.tex @@ -4,7 +4,7 @@ \subsubsection{Construction of iid random variables.} \begin{itemize} - \item Definition of a consistent family (\autoref{def:consistentfamily}) + \item Definition of a consistent family (\yaref{def:consistentfamily}) \item Important construction: Consider a distribution function $F$ and define @@ -16,22 +16,22 @@ \item Examples of consistent and inconsistent families \todo{Exercises} \item Kolmogorov's consistency theorem - (\autoref{thm:kolmogorovconsistency}) + (\yaref{thm:kolmogorovconsistency}) \end{itemize} \subsubsection{Limit theorems} \begin{itemize} \item Work with iid.~random variables. - \item Notions of convergence (\autoref{def:convergence}) + \item Notions of convergence (\yaref{def:convergence}) \item Implications between different notions of convergence (very important) and counter examples. - (\autoref{thm:convergenceimplications}) + (\yaref{thm:convergenceimplications}) - \item Laws of large numbers: (\autoref{lln}) + \item Laws of large numbers: (\yaref{lln}) \begin{itemize} \item WLLN: convergence in probability \item SLLN: weak convergence \end{itemize} - \item \autoref{thm2} (building block for SLLN): + \item \yaref{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} @@ -43,12 +43,12 @@ $\sum \frac{\pm 1}{ n^{\frac{1}{2} -\epsilon}}$ 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}) + \item \yaref{thm:kolmogorovineq} + \item \yaref{kolmogorov01} In particular, a series of independent random variables converges with probability $0$ or $1$. - \item Kolmogorov's 3 series theorem. (\autoref{thm:kolmogorovthreeseries}) + \item \yaref{thm:kolmogorovthreeseries} \begin{itemize} \item What are those $3$ series? \item Applications @@ -59,15 +59,15 @@ \begin{itemize} \item Definition of Fourier transform - (\autoref{def:characteristicfunction}) + (\yaref{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 Bochner's theorem for positive definite function (\autoref{thm:bochner}) - \item Bochner's theorem for the mass at a point (\autoref{bochnersformula}) + \item \yaref{charfuncuniqueness}, + \yaref{inversionformula}, ... + \item \yaref{levycontinuity}, + \yaref{genlevycontinuity} + \item \yaref{thm:bochner} + \item \yaref{bochnersformula} \item Related notions \todo{TODO} \begin{itemize} @@ -81,7 +81,7 @@ \paragraph{Weak convergence} \begin{itemize} \item Definition of weak convergence % ( test against continuous, bounded functions). - (\autoref{def:weakconvergence}) + (\yaref{def:weakconvergence}) \item Examples: \begin{itemize} \item $(\delta_{\frac{1}{n}})_n$, @@ -93,9 +93,8 @@ \item Non-examples: $(\delta_n)_n$ \item How does one prove weak convergence? How does one write this down in a clear way? \begin{itemize} - \item \autoref{lec10_thm1}, - \item Levy's continuity theorem - \ref{levycontinuity}, + \item \yaref{lec10_thm1}, + \item \yaref{levycontinuity}, \item Generalization of Levy's continuity theorem \ref{genlevycontinuity} \end{itemize} @@ -111,12 +110,12 @@ \subsubsubsection{CLT} \begin{itemize} - \item Statement of the CLT + \item Statement of the \yaref{clt} \item Several versions: \begin{itemize} - \item iid (\autoref{clt}), - \item Lindeberg (\autoref{lindebergclt}), - \item Lyapanov (\autoref{lyapunovclt}) + \item iid, + \item \yaref{lindebergclt}, + \item \yaref{lyapunovclt} \end{itemize} \item How to apply this? Exercises! \end{itemize} @@ -124,11 +123,11 @@ \subsubsection{Conditional expectation} \begin{itemize} \item Definition and existence of conditional expectation for $X \in L^1(\Omega, \cF, \bP)$ - (\autoref{conditionalexpectation}) + (\yaref{conditionalexpectation}) \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 \ref{radonnikodym} + \item \yaref{radonnikodym} (Proof not relevant for the exam) \item (Non-)examples of mutually absolutely continuous measures Singularity in this context? % TODO @@ -137,30 +136,30 @@ \subsubsection{Martingales} \begin{itemize} - \item Definition of Martingales (\autoref{def:martingale}) - \item Doob's convergence theorem (\autoref{doobmartingaleconvergence}), - Upcrossing inequality (\autoref{lec17l1}, \autoref{lec17l2}, \autoref{lec17l3}) + \item Definition of Martingales (\yaref{def:martingale}) + \item Doob's convergence theorem (\yaref{doobmartingaleconvergence}), + Upcrossing inequality (\yaref{lec17l1}, \yaref{lec17l2}, \yaref{lec17l3}) (downcrossings for submartingales) \item Examples of Martingales converging a.s.~but not in $L^1$ - (\autoref{ex:martingale-not-converging-in-l1}) + (\yaref{ex:martingale-not-converging-in-l1}) \item Bounded in $L^2$ $\implies$ convergence in $L^2$ - (\autoref{martingaleconvergencel2}). + (\yaref{martingaleconvergencel2}). \item Martingale increments are orthogonal in $L^2$! - (\autoref{martingaleincrementsorthogonal}) + (\yaref{martingaleincrementsorthogonal}) \item Doob's (sub-)martingale inequalities - (\autoref{dooblp}), + (\yaref{dooblp}), \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 - (\autoref{ceismartingale}, \autoref{martingaleisce}). + (\yaref{ceismartingale}, \yaref{martingaleisce}). \begin{itemize} - \item Why is $p > 1$ important? \textbf{Role of Banach-Alaoglu} + \item Why is $p > 1$ important? \textbf{Role of \yaref{banachalaoglu}} \item This is an important proof. \end{itemize} - \item Uniform integrability (\autoref{def:ui}) - \item What are stopping times? (\autoref{def:stopping-time}) + \item Uniform integrability (\yaref{def:ui}) + \item What are stopping times? (\yaref{def:stopping-time}) \item (Non-)examples of stopping times - \item \textbf{Optional stopping theorem} (\autoref{optionalstopping}) + \item \textbf{\yaref{optionalstopping}} - be really comfortable with this. \end{itemize} diff --git a/inputs/prerequisites.tex b/inputs/prerequisites.tex index 51994a0..32199fb 100644 --- a/inputs/prerequisites.tex +++ b/inputs/prerequisites.tex @@ -35,7 +35,7 @@ from the lecture on stochastic. This notion of convergence was actually defined during the course of the lecture, but has been added here for completeness; - see \autoref{def:weakconvergence}. + see \yaref{def:weakconvergence}. } ($X_n \xrightarrow{\text{d}} X$) iff for every continuous, bounded $f: \R \to \R$ @@ -128,7 +128,7 @@ from the lecture on stochastic. \begin{subproof} Let $F$ be the distribution function of $X$ and $(F_n)_n$ the distribution functions of $(X_n)_n$. - By \autoref{lec10_thm1} + By \yaref{lec10_thm1} it suffices to show that $F_n(t) \to F(t)$ for all continuity points $t$ of $F$. Let $t$ be a continuity point of $F$. @@ -155,7 +155,7 @@ from the lecture on stochastic. \label{claim:convimplpl1} $X_n \xrightarrow{\bP} X \notimplies X_n\xrightarrow{L^1} X$.% \footnote{Note that the implication holds under certain assumptions, - see \autoref{thm:l1iffuip}.} + see \yaref{thm:l1iffuip}.} \end{claim} \begin{subproof} Take $([0,1], \cB([0,1 ]), \lambda)$ @@ -170,7 +170,7 @@ from the lecture on stochastic. $X_n \xrightarrow{\text{a.s.}} X \notimplies X_n\xrightarrow{L^1} X$. \end{claim} \begin{subproof} - We can use the same counterexample as in \autoref{claim:convimplpl1} + We can use the same counterexample as in \yaref{claim:convimplpl1} $\bP[\lim_{n \to \infty} X_n = 0] \ge \bP[X_n = 0] = 1 - \frac{1}{n} \to 0$. We have already seen, that $X_n$ does not converge in $L_1$. @@ -220,7 +220,7 @@ from the lecture on stochastic. \end{refproof} \begin{theorem}[Bounded convergence theorem] - \label{thm:boundedconvergence} + \yalabel{Bounded Convergence Theorem}{Bounded convergence}{thm:boundedconvergence} Suppose that $X_n \xrightarrow{\bP} X$ and there exists some $K$ such that $|X_n| \le K$ for all $n$. Then $X_n \xrightarrow{L^1} X$. @@ -262,7 +262,7 @@ from the lecture on stochastic. \begin{theorem}+[Riemann-Lebesgue] %\footnote{see exercise 3.3} - \label{riemann-lebesgue} + \yalabel{Riemann-Lebesgue}{Riemann-Lebesgue}{riemann-lebesgue} Let $f: \R \to \R$ be integrable. Then \[ @@ -271,6 +271,7 @@ from the lecture on stochastic. \end{theorem} \begin{theorem}+[Fubini-Tonelli] + \yalabel{Fubuni-Tonelli}{Fubini}{thm:fubini} %\footnote{exercise sheet 1} Let $(\Omega_{i}, \cF_i, \bP_i), i \in \{0,1\}$ be probability spaces and $\Omega \coloneqq \Omega_0 \otimes \Omega_1$, @@ -296,6 +297,7 @@ from the lecture on stochastic. This is taken from section 6.1 of the notes on Stochastik. \begin{theorem}[Markov's inequality] + \yalabel{Markov's Inequality}{Markov}{thm:markov} Let $X$ be a random variable and $a > 0$. Then \[ @@ -311,6 +313,7 @@ This is taken from section 6.1 of the notes on Stochastik. \end{proof} \begin{theorem}[Chebyshev's inequality] + \yalabel{Chebyshev's Inequality}{Chebyshev}{thm:chebyshev} Let $X$ be a random variable and $a > 0$. Then \[ @@ -322,14 +325,14 @@ This is taken from section 6.1 of the notes on Stochastik. \begin{IEEEeqnarray*}{rCl} \bP[|X-\bE(X)| \ge a] &=& \bP[|X - \bE(X)|^2 \ge a^2]\\ - &\overset{\text{Markov}}{\le}& \frac{\bE[|X - \bE(X)|^2]}{a^2}. + &\overset{\yaref{thm:markov}}{\le}& \frac{\bE[|X - \bE(X)|^2]}{a^2}. \end{IEEEeqnarray*} \end{proof} How do we prove that something happens almost surely? The first thing that should come to mind is: \begin{lemma}[Borel-Cantelli] - \label{borelcantelli} + \yalabel{Borel-Cantelli}{Borel-Cantelli}{thm:borelcantelli} If we have a sequence of events $(A_n)_{n \ge 1}$ such that $\sum_{n \ge 1} \bP(A_n) < \infty$, then $\bP[ A_n \text{for infinitely many $n$}] = 0$ diff --git a/jrpie-yaref.sty b/jrpie-yaref.sty new file mode 100644 index 0000000..4fd9b6a --- /dev/null +++ b/jrpie-yaref.sty @@ -0,0 +1,49 @@ +\NeedsTeXFormat{LaTeX2e} +\ProvidesPackage{jrpie-yaref}[2023/07/28 - yet another ref] + +\RequirePackage{hyperref} +\RequirePackage{amstext} + +\newcommand{\yaref@text@large}[1]{% + \ifcsname yaref@longlabel@#1\endcsname% + \hyperref[#1]{\csname yaref@longlabel@#1\endcsname\ \ref*{#1}}% + \else% + \autoref{#1}% + \fi% +} +\newcommand{\yaref@text@small}[1]{% + \ifcsname yaref@shortlabel@#1\endcsname% + \hyperref[#1]{\csname yaref@shortlabel@#1\endcsname}% + \else% + (\ref{#1})% + \fi% +} +\newcommand{\yaref@math@large}[1]{% + \text{\yaref@text@large{#1}}% +} +\newcommand{\yaref@math@small}[1]{% + \text{\yaref@text@small{#1}}% +} +\newcommand{\yaref@math@verysmall}[1]{% + \yaref@math@small{#1}% +} + +\newcommand{\yalabel}[3]{% + \write\@auxout{\noexpand\expandafter\noexpand\gdef\noexpand\csname yaref@longlabel@#3\noexpand\endcsname{#1}}% + \write\@auxout{\noexpand\expandafter\noexpand\gdef\noexpand\csname yaref@shortlabel@#3\noexpand\endcsname{#2}}% + \expandafter\gdef\csname yaref@longlabel@#3\endcsname{#1}% + \expandafter\gdef\csname yaref@shortlabel@#3\endcsname{#2}% + \label{#3}% +} + +\newcommand{\yaref}[1]{% + \relax\ifmmode% + \mathchoice + {\yaref@math@large{#1}} % display style + {\yaref@math@large{#1}} % text style + {\yaref@math@small{#1}} % script style + {\yaref@math@verysmall{#1}} % scriptscript style + \else% + \yaref@text@large{#1}% + \fi% +} diff --git a/wtheo.sty b/wtheo.sty index 852d865..3b00e6f 100644 --- a/wtheo.sty +++ b/wtheo.sty @@ -8,6 +8,7 @@ \usepackage[index]{mkessler-vocab} \usepackage{mkessler-code} \usepackage{jrpie-math} +\usepackage{jrpie-yaref} \usepackage[normalem]{ulem} \usepackage{pdflscape} \usepackage{longtable}