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\documentclass{amsart}
\title{Digest of Chung (2003), Chapter 5}
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\begin{document}
\begin{CJK*}{UTF8}{cwku}
\author{國立中興大學\hspace{.5em}應用數學系\hspace}
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\section{Simple limit theorems}
The {\em weak law of largenumbers} is said to hold for the sequence
according as \begin{equation}\label{wlln}\frac{S_n -\mathscr
E(S_n)}{n}\rightarrow 0
\end{equation}
in pr. or a.e. A natural generalization is as follow
\begin{equation*} \frac{S_n - a_n}{b_n}\rightarrow 0
\end{equation*}
\begin{thm}
If the $X_j$'s are uncorrelated and their second moments have a
common bound, then \eqref{wlln} is true in $L^2$ and hence also in
pr. and a.e..
\end{thm}
\begin{proof}
I just fill the gap left on Chung's text book. Chung gave the result
that $\mathscr P\{D_n > n^2\epsilon\}\leq \frac{4M}{\epsilon^2n^2}$,
thus, \[\sum_n \mathscr P\{D_n > n^2\epsilon\} \leq \sum_n
\frac{4M}{\epsilon^2n^2} =
\frac{4M}{\epsilon^2n}\overset{n\rightarrow\infty}{\rightarrow} 0\]
thus, by Borel-Cantelli's lemma, $\mathscr P\{D_n > n^2\epsilon
\hspace{.5em}\text{i.o.} \} = 0$, thus, $\frac{D_n}{n^2}\rightarrow 0$ a.e.
\[\frac{|S_k|}{k} = \frac{|S_k - S_{n^2} + S_{n^2} |}{k}
\leq\frac{|S_k - S_{n^2}| + |S_{n^2} |}{k}\leq\frac{|S_k - S_{n^2}| +
|S_{n^2} |}{n^2}\] for $n^2\leq k\leq(n+1)^2$
\end{proof}
\section{Weak law of largenumbers}
\section{Convergence of series}
\section{Strong law of largenumbers}
\section{Applications}
\end{document}
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