From 9f698ddf03153e53dffd7e08b2f3825498eeafed Mon Sep 17 00:00:00 2001 From: Josia Pietsch Date: Wed, 12 Jul 2023 15:27:13 +0200 Subject: [PATCH] overfull hbox --- inputs/lecture_06.tex | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/inputs/lecture_06.tex b/inputs/lecture_06.tex index a2adaad..414310b 100644 --- a/inputs/lecture_06.tex +++ b/inputs/lecture_06.tex @@ -63,9 +63,9 @@ In order to prove \autoref{thm2}, we need the following: |X_1(\omega) + X_2(\omega)| > \epsilon \},\\ \ldots\\ A_i &\coloneqq& \{\omega: |X_1(\omega)| \le \epsilon, - |X_1(\omega) + X_2(\omega)| \le \epsilon, \ldots, - |X_1(\omega) + \ldots + X_{i-1}(\omega)| \le \epsilon, - |X_1(\omega) + \ldots + X_i(\omega)| > \epsilon\}. + |X_1(\omega) + X_2(\omega)| \le \epsilon, \ldots, % + |X_1(\omega) + \ldots + X_{i-1}(\omega)| \le \epsilon,\\ + && ~ ~|X_1(\omega) + \ldots + X_i(\omega)| > \epsilon\}. \end{IEEEeqnarray*} It is clear, that the $A_i$ are disjoint. We are interested in $\bigcup_{1 \le i \le n} A_i$. @@ -179,7 +179,7 @@ In order to prove \autoref{thm2}, we need the following: since $\{N_t \ge n\} = \{X_1 + \ldots+ X_n \le t\}$. \begin{claim} - $\bP[\frac{S_n}{n} \xrightarrow{n \to \infty} m + $\bP[\frac{S_n}{n} \xrightarrow{n \to \infty} m \land N_t \xrightarrow{t \to \infty} \infty] = 1$. \end{claim} \begin{subproof}