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研究生:陳琮威
研究生(外文):Tsung-Wei Chen
論文名稱:Klesov定理中收斂速率之研究
論文名稱(外文):On the Convergence Rate in a Theorem of Klesov
指導教授:蔡志賢蔡志賢引用關係
指導教授(外文):Jhishen Tsay
學位類別:碩士
校院名稱:國立中山大學
系所名稱:應用數學系研究所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2004
畢業學年度:92
語文別:英文
論文頁數:12
中文關鍵詞:完備收斂收斂速度
外文關鍵詞:convergence ratecomplete convergence
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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
1. P. L. Hsu and H. Robbins (1947), Complete convergence and the law of large numbers , Proc. Nat. Acad. Sci. U. S. A. 33 , no. 2,25-31.

2. P. Erd$ddot{o}$s (1949), On a theorem of Hsu and Robbins
, Ann. Math. Statist. 20, no. 2, 286-291.

3. P. Erd$ddot{o}$s (1950), Remark on my paper $"$On a theorem of Hsu and Robbins$"$ , Ann. Math. Statist. 21, no. 1, 138-138.

4. Bikjalis, A. (1966), Estimates of the remainder term in the central limit theorem, Litovsk. Mat. Sb.6.323-346.

5. John Slivka and N.C. Severo (1970), On the strong law of large numbers, Proc. Amer. Math. Soc. 24, 729-734.

6. C. F. Wu (1973), A note on the convergence rate of the strong law of
large numbers, Bull. Inst. Math. Acad. Sinica. 1, 121-124.

7. C. C. Heyde (1975), A supplement to the strong law of large numbers, J. Appl. Probab. 12, no. 1, 903-907.

8. Robert Chen (1976), A remark on the strong law of large numbers,Proc. Amer. Math. Soc, Vol. 61, no. 1, 112-116.

9. R. G. Laha and V.
K. Rohatgi, Probability Theory, Wiley, New York, 1979.

10.R. N. Bhattacharya and R. Ranga Rao, Normal approximation and asymptotic expansions, 2nd ed., 1986.

11. O.I. Klesov (1994), On the convergence rate in a theorem of Heyde, Theor. Probab. and Math. Statist. no. 49, 83-87.
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