設${X_{n}}^{infty}_{n=1}$為獨立且具有共同分佈之隨機變數，令$S_{n}=displaystyle
sum^{n}_{k=1}X_{k}$。定義 $lambda(varepsilon)=displaystyle sum^{infty}_{n=1}P{|S_{n}|geq
nvarepsilon}$，本文主要來探討 $lambda(varepsilon)$
的收斂速度。O.I. Klesove 指出如果 $EX_{1}=0, EX_{1}^{2}=
ho^{2}
eq 0, E|X_{1}|^{3}<infty$，
則 $displaystylelim_{varepsilondownarrow0}displaystylevarepsilon^{frac{3}{2}}(lambda(varepsilon)-frac{sigma^{2}}{varepsilon^{2}})=0$，
這篇論文則是證明如果條件 $E|X_{1}|^{3}<infty$ 設為
$displaystyleforalldeltain(frac{1}{2},1]$，
$E|X_{1}|^{2+delta}<infty$， 則
$displaystylelim_{varepsilondownarrow0}displaystylevarepsilon^{frac{3}{2}}(lambda(varepsilon)-frac{sigma^{2}}{varepsilon^{2}})=0$
亦會成立 。

egin{abstract}
hspace*{1cm} Let $X_{1}$@, $X_{2}$@,$cdots$@, $X_{n}$ be a sequence of
independent
indentically distributed random variables ( i@. i@. d@.) and
$S_{n}=X_{1}+X_{2}+...+X_{n}$@. Denote
$lambda(varepsilon)=displaystylesum_{n=1}^{infty}P(|S_{n}|geq
nvarepsilon)$@. O.I. Klesov proved that if $EX_{1}=0$,
$EX_{1}^{2}=sigma ^{2}
eq 0$, $E|X_{1}|^{3}<infty$, then
$displaystylelim_{varepsilondownarrow0}varepsilon^{frac{3}{2}}(lambda(varepsilon)-frac{sigma^{2}}{varepsilon^{2}})=0$.
In this thesis, it is shown that if $EX_{1}=0$,
$EX_{1}^{2}=sigma ^{2}
eq 0$, $E|X_{1}|^{2+delta}<infty$ for
some $displaystyledeltain(frac{1}{2},1]$, then
$displaystylelim_{varepsilondownarrow0}varepsilon^{frac{3}{2}}(lambda(varepsilon)-frac{sigma^{2}}{varepsilon^{2}})=0$.
end{abstract}

1. Introduction -------------1
2. Main result -------------5
3. Reference -------------11

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