From 594d933beb704d574ab817c6c627ab643b19a290 Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Thu, 13 Jul 2023 18:35:33 +0200 Subject: [PATCH] fixed typo --- inputs/prerequisites.tex | 9 +++++---- 1 file changed, 5 insertions(+), 4 deletions(-) diff --git a/inputs/prerequisites.tex b/inputs/prerequisites.tex index 6c825bf..ceb9c15 100644 --- a/inputs/prerequisites.tex +++ b/inputs/prerequisites.tex @@ -77,16 +77,17 @@ from the lecture on stochastic. $X_n \xrightarrow{\text{a.s.}} X \implies X_n \xrightarrow{\bP} X$. \end{claim} \begin{subproof} - $\Omega_0 \coloneqq + Let $\Omega_0 \coloneqq \{\omega \in \Omega : \lim_{n\to \infty} X_n(\omega) = X(\omega)\}$. - Let $\epsilon > 0$ and consider + Fix some $\epsilon > 0$ and consider $A_n \coloneqq \bigcup_{m \ge n} \{\omega \in \Omega: |X_m(\omega) - X(\omega)| > \epsilon\}$. Then $A_n \supseteq A_{n+1} \supseteq \ldots$ Define $A \coloneqq \bigcap_{n \in \N} A_n$. Then $\bP[A_n] \xrightarrow{n\to \infty} \bP[A]$. Since $X_n \xrightarrow{a.s.} X$ we have that - $\forall \omega \in \Omega_0 \exists n \in \N \forall m \ge n |X_m(\omega) - X(\omega)| < \epsilon$. + \[\forall \omega \in \Omega_0 .~ \exists n \in \N .~ + \forall m \ge n.~ |X_m(\omega) - X(\omega)| < \epsilon.\] We have $A \subseteq \Omega_0^{c}$, hence $\bP[A_n] \to 0$. Thus \[ \bP[\{\omega \in \Omega | ~|X_n(\omega) - X(\omega)| > \epsilon\}] < \bP[A_n] \to 0. @@ -114,7 +115,7 @@ from the lecture on stochastic. Then for every $\epsilon > 0$ \begin{IEEEeqnarray*}{rCl} \bP[|X_n - X| \ge \epsilon] - &\overset{\text{Markov}}{\ge}& \frac{\bE[|X_n - X|]}{\epsilon}\\ + &\overset{\text{Markov}}{\le}& \frac{\bE[|X_n - X|]}{\epsilon}\\ &\xrightarrow{n \to \infty} & 0, \end{IEEEeqnarray*} hence $X_n \xrightarrow{\bP} X$.