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姓名 黃彥青( Yan-Qing Huang)  查詢紙本館藏   畢業系所 數學系
論文名稱 一些關於雙獨立序列的機率收斂定理
(PAIRWISE INDEPENDENT RANDOM VARIABLES)
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摘要(中) Janson 在1988年提出了反例,證明對於一個雙獨立隨機變數序列,中央極限定理是不成立的。我們進一步問:是否可以加上一些條件,使得中央極限定理成立? 也就是說,我們想找出使雙獨立隨機變數序列具有中央極限定理的充分條件。
McLeish 在1974發表了一篇論文:”Dependent central limit theorem and invariance principles “,其中的定理2.1就提供了一個答案。這個定理有四個條件,其中兩個是針對所考慮的隨機變數序列本身,另外兩個是針對由此一隨機變數序列而定出的函數序列。但是這四個條件也不容易檢驗,我們希望可以找到比較容易檢驗而且可以推導出這四個條件的條件,那麼就可以取得實際運用上較大的便利。
本文假設其中兩個針對函數序列 (由所考慮的隨機變數序列而定出) 要求的條件成立,全力在另外兩個跟機率收斂有關的條件下工夫。先依Chandra所提出的Cesaro 均勻可積得到兩個定理;再依另一個由Hong提出,較Cesaro 均勻可積弱的條件得到其他的定理。
最後舉出一個雙獨立隨機變數序列作為例證。
摘要(英) hold. In my thesis, some related results are mentioned. I also give
some new version of conditions such that the central limit theorem would
hold for pairwise independent sequences. Finally, I give an example to
illustrate the results.
關鍵字(中) ★ 雙獨立隨機變數序列
★ 相互獨立
★ 中央極限定理
★ 隨機變數序列
關鍵字(英) ★ some related results are mentioned
論文目次 Contents
1. Introduction 1
2. The results based on u.i.c. 4
3. The main results 11
4. Example 17
Reference 20
參考文獻 [1] A. Bose and T. K. Chandra, Cesaro uniform integrability and Lp- convergence,
Sankhya Ser. A 55 (1993), 12-28.
[2] R. C. Bradley, A stationary, pairwise independent, absolutely regular sequence for
which the Central Limit Theorem fails, Theory of Probab. and Related Fields 81
(1989), 1-10.
[3] S. M. Berman, Sign-invariant random variables and stochastic processes with
sign- invariant increments, Trans. Amer. Math. Soc. 119 (1965), 216-243.
[4] J. A. Cuesta, C. Matran, On the asymptotic behavior of sums of pairwise in-
dependent random variables, Statist. Probab. Lett. 11 (1991), 201-210.
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large numbers, Sankhya Ser. A 51 (1989), 309-317.
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20
[15] S. Janson, Some pairwise independent sequences for which the Central Limit
Theorem fails, Stochastics 23 (1988), 439-448.
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which the central limit theorem holds, Yokohama Math. J. 45 (1998), 87-96.
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dent random variables, Statist. Probab. Lett. 25 (1995), 21-26.
[19] D. L. McLeish, Dependent central limit theorem and invariance principles, Ann.
Probab. 2 (1974), 620-628.
[20] D. L. McLeish , On the invariance principle for nonstationary mixingales, Ann.
Probab. 5 (1978), 616-621.
[21] J. B. Robertson and J. B. Womak, A pairwise independent stationary stochastic
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[22] P. Schatte, Note on the concept of m-dependence, Math. Nachr. 140 (1989), 249-
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[23] S. H. Sung, Weak law of large numbers for arrays, Statist. Probab. Lett. 38 (1998),
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[24] M. S. Taqqu, Convergence of integrated processes of arbitrary Hermite rank, Z.
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指導教授 周元燊(Yuan-Shen Chou) 審核日期 2001-6-25
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