The Annals of Probability

On the Glivenko-Cantelli Theorem for Weighted Empiricals Based on Independent Random Variables

Radhey S. Singh

Full-text: Open access

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 $O((n \log n)^{\frac{1}{2}})$.

Article information

Source
Ann. Probab., Volume 3, Number 2 (1975), 371-374.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176996410

Digital Object Identifier
doi:10.1214/aop/1176996410

Mathematical Reviews number (MathSciNet)
MR372971

Zentralblatt MATH identifier
0312.60014

JSTOR
links.jstor.org

Subjects
Primary: 60F10: Large deviations
Secondary: 60F15: Strong theorems

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

Citation

Singh, Radhey S. On the Glivenko-Cantelli Theorem for Weighted Empiricals Based on Independent Random Variables. Ann. Probab. 3 (1975), no. 2, 371--374. doi:10.1214/aop/1176996410. https://projecteuclid.org/euclid.aop/1176996410


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