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April, 1975 On the Glivenko-Cantelli Theorem for Weighted Empiricals Based on Independent Random Variables
Radhey S. Singh
Ann. Probab. 3(2): 371-374 (April, 1975). DOI: 10.1214/aop/1176996410

Abstract

For $X_1, \cdots, X_n$ independent real valued random variables and for $\alpha \in \lbrack 0, 1 \rbrack$, let $F_j(x) = \alpha P\lbrack X_j < x \rbrack + (1 - \alpha)P\lbrack X_j \leqq x \rbrack$ and $Y_j(x) = \alpha I_{\lbrack X_j < x \rbrack} + (1 - \alpha) I_{\lbrack X_j \leqq x \rbrack}$, where $I_A$ is the indicator function of the set $A$. For numbers $w_1, w_2, \cdots, w_n$, let $D_n = \sup_{x, \alpha} \max_{N \leqq n}|\sum^N_1 w_j(Y_j(x) - F_j(x))|$. We will obtain an exponential bound for $P\lbrack D_n \geqq a \rbrack$ and a rate for almost sure convergence of $D_n$. When $w_j \equiv 1$ the bound and the rate become, respectively, $4a \exp \{-2((a^2/n) - 1)\}$ and $O((n \log n)^{\frac{1}{2}})$.

Citation

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Radhey S. Singh. "On the Glivenko-Cantelli Theorem for Weighted Empiricals Based on Independent Random Variables." Ann. Probab. 3 (2) 371 - 374, April, 1975. https://doi.org/10.1214/aop/1176996410

Information

Published: April, 1975
First available in Project Euclid: 19 April 2007

zbMATH: 0312.60014
MathSciNet: MR372971
Digital Object Identifier: 10.1214/aop/1176996410

Subjects:
Primary: 60F10
Secondary: 60F15

Keywords: Borel-Cantelli lemma , Glivenko-Cantelli theorem , independent non-identically distributed , weighted empiricals

Rights: Copyright © 1975 Institute of Mathematical Statistics

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Vol.3 • No. 2 • April, 1975
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