| Almost sure central limit theorems for random functions |
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| 作者姓名： | LU Chuanrong QIU Jin & XU Jianjun School of Mathematics and Statistics Zhejiang University of Finance and Economics Hangzhou China |
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| 作者单位： | LU Chuanrong，QIU Jin & XU Jianjun School of Mathematics and Statistics，Zhejiang University of Finance and Economics，Hangzhou 310018，China; Department of Mathematics，Zhejiang University，Hangzhou 310028，China |
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| 基金项目： | 浙江省自然科学基金;国家自然科学基金 |
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| 摘 要： | Let {Xn,-∞< n <∞} be a sequence of independent identically distributed random variables with EX1 = 0, EX12 = 1 and let Sn =∑k=1∞Xk, and Tn = Tn(X1,…,Xn) be a random function such that Tn = ASn Rn, where supn E|Rn| <∞and Rn = o(n~(1/2)) a.s., or Rn = O(n1/2-2γ) a.s., 0 <γ< 1/8. In this paper, we prove the almost sure central limit theorem (ASCLT) and the function-typed almost sure central limit theorem (FASCLT) for the random function Tn. As a consequence, it can be shown that ASCLT and FASCLT also hold for U-statistics, Von-Mises statistics, linear processes, moving average processes, error variance estimates in linear models, power sums, product-limit estimators of a continuous distribution, product-limit estimators of a quantile function, etc.
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| 收稿时间： | 16 March 2005 |
| 修稿时间： | 26 June 2006 |
Almost sure central limit theorems for random functions |
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| LU Chuanrong,QIU Jin & XU Jianjun School of Mathematics and Statistics,Zhejiang University of Finance and Economics,Hangzhou ,China,.Almost sure central limit theorems for random functions[J].Science in China(Mathematics),2006,49(12):1788-1799. |
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| Authors: | LU Chuanrong QIU Jin XU Jianjun |
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| Institution: | 1. School of Mathematics and Statistics,Zhejiang University of Finance and Economics,Hangzhou 310018,China;Department of Mathematics,Zhejiang University,Hangzhou 310028,China
2. Department of Mathematics,Zhejiang University,Hangzhou 310028,China |
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| Abstract: | Let {X
n, −∞ < n < ∞} be a sequence of independent identically distributed random variables with EX
1 = 0, EX
1
2
= 1 and let S
n = Σ
k=1
∞
X
k, and T
n = T
n(X
1, ..., X
n) be a random function such that T
n = AS
n + R
n, where supn E|R
n| < ∞ and R
n = o(hrn) a.s., or R
n = O(n
1/2−2γ) a.s., 0 < γ < 1/8. In this paper, we prove the almost sure central limit theorem (ASCLT) and the function-typed almost sure
central limit theorem (FASCLT) for the random function T
n. As a consequence, it can be shown that ASCLT and FASCLT also hold for U-statistics, Von-Mises statistics, linear processes,
moving average processes, error variance estimates in linear models, power sums, product-limit estimators of a continuous
distribution, product-limit estimators of a quantile function, etc. |
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| Keywords: | statistics random function almost sure central limit theorem logarithm average |
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