From 9c517a05b2b1fc59ca1b9746e1147e8789e5416f Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Sat, 15 Jul 2023 00:57:02 +0200 Subject: [PATCH] Bounded convergence theorem --- inputs/lecture_19.tex | 12 ++++++------ inputs/lecture_20.tex | 5 +++-- inputs/prerequisites.tex | 17 +++++++++++++++++ 3 files changed, 26 insertions(+), 8 deletions(-) diff --git a/inputs/lecture_19.tex b/inputs/lecture_19.tex index 1a9ce99..366e43f 100644 --- a/inputs/lecture_19.tex +++ b/inputs/lecture_19.tex @@ -136,10 +136,10 @@ However, some subsets can be easily described, e.g. Let $Y = \bE[X | \cG]$ for some sub-$\sigma$-algebra $\cG$. Then, by \autoref{cjensen}, $|Y| \le \bE[ |X| | \cG]$. Hence $\bE[|Y|] \le \bE[|X|]$. - It follows that $\bP[|Y| > k] < \delta$ - for $k$ suitably large, - since $\bE[|X|] \le \infty$. - Note that $\{Y > k\} \in \cG$. + By Markov's inequality, + it follows that $\bP[|Y| > k] < \delta$ + for $k > \frac{\bE[|X|]}{\delta}$. + Note that $\{|Y| > k\} \in \cG$. We have \begin{IEEEeqnarray*}{rCl} \bE[|Y| \One_{\{|Y| > k\} }] &<& \epsilon @@ -200,7 +200,7 @@ However, some subsets can be easily described, e.g. \end{IEEEeqnarray*} for all $\delta > 0$ and suitable $k$. - Hence $\bP[|X_n| \ge k] < \delta$ by Markov's inequality. + Hence $\bP[|X_n| > k] < \delta$ by Markov's inequality. Then by \autoref{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. @@ -232,7 +232,7 @@ Let $(\Omega, \cF, \bP)$ as always and let $(\cF_n)_n$ always be a filtration. \begin{theorem} \label{ceismartingale} Let $X \in L^p$ for some $p \ge 1$ - and $\bigcup_n \cF_n \to \cF$. + and $\bigcup_n \cF_n \to \cF \supseteq \sigma(X)$. Then $X_n \coloneqq \bE[X | \cF_n]$ defines a martingale which converges to $X$ in $L^p$. \end{theorem} diff --git a/inputs/lecture_20.tex b/inputs/lecture_20.tex index 0d15cc7..154db22 100644 --- a/inputs/lecture_20.tex +++ b/inputs/lecture_20.tex @@ -45,7 +45,8 @@ \int | X - X'|^p \dif \bP &=& \int_{\{|X| > M\} } |X|^p \dif \bP \xrightarrow{M \to \infty} 0 \end{IEEEeqnarray*} as $\bP$ is regular, - i.e.~$\forall \epsilon > 0 . ~\exists k . ~\bP[|X|^p \in [-k,k] \ge 1-\epsilon$. + i.e.~$\forall \epsilon > 0 . ~\exists k . ~ + \bP[|X|^p \in [-k,k]] \ge 1-\epsilon$. Take some $\epsilon > 0$ and $M$ large enough such that \[ @@ -111,7 +112,7 @@ we need the following theorem, which we won't prove here: \int_A X \dif \bP &=& \lim_{k \to \infty} \int_A X_{n_k} \dif \bP\\ &=& \lim_{k \to \infty} \bE[X_{n_k} \One_A]\\ - &\overset{\text{for }n_k \ge m}{=}& \int_{k \to \infty} \bE[X_m \One_A]. + &\overset{\text{for }n_k \ge m}{=}& \lim_{k \to \infty} \bE[X_m \One_A]. \end{IEEEeqnarray*} Hence $X_n = \bE[X | \cF_m]$ by the uniqueness of conditional expectation and by \autoref{ceismartingale}, diff --git a/inputs/prerequisites.tex b/inputs/prerequisites.tex index ceb9c15..9424694 100644 --- a/inputs/prerequisites.tex +++ b/inputs/prerequisites.tex @@ -215,6 +215,23 @@ from the lecture on stochastic. \end{subproof} \end{refproof} +\begin{theorem}[Bounded convergence theorem] + \label{thm:boundedconvergence} + Suppose that $X_n \xrightarrow{\bP} X$ and there exists + some $K$ such that $|X_n| \le K$ for all $n$. + Then $X_n \xrightarrow{L^1} X$. +\end{theorem} +\begin{proof} + Note that $|X| \le K$ a.s.~since + \[\bP[|X| \ge K + \epsilon] \le \bP[|X_n - X| > \epsilon] + \xrightarrow{n \to \infty} 0.\] + Hence + \begin{IEEEeqnarray*}{rCl} + \int |X_n - X| \dif \bP &\le& \int_{|X_n - X| \ge \epsilon} |X_n - X| \dif \bP + \epsilon\\ + &\le &2K\bP[|X_n - X| \ge \epsilon] +\epsilon. + \end{IEEEeqnarray*} +\end{proof} + \subsection{Some Facts from Measure Theory} \begin{fact}+[Finite measures are {\vocab[Measure]{regular}}, Exercise 3.1]