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October, 1975 A Functional Law of the Iterated Logarithm for Weighted Empirical Distributions
Barry R. James
Ann. Probab. 3(5): 762-772 (October, 1975). DOI: 10.1214/aop/1176996263

Abstract

Finkelstein's (1971) functional law of the iterated logarithm for empirical distributions is extended to cases where the empirical distribution is multiplied by a weight function, $w$. We let $X_1, X_2, \cdots$ be independent random variables each having the uniform distribution on $\lbrack 0, 1 \rbrack$, with $F_n$ the empirical df at stage $n$. The weight function $w$, defined on $\lbrack 0, 1 \rbrack$, is assumed to be bounded on interior intervals and to satisfy some smoothness conditions. Then convergence of the integral $\int^1_0 w^2(t)/\log \log(t^{-1}(1 - t)^{-1})dt$ is seen to be a necessary and sufficient condition for the sequence $\{U_n: n \geqq 3\}$, defined by $$U_n(t) = \frac{n^{\frac{1}{2}}w(t)(F_n(t) - t)}{(2 \log \log n)^{\frac{1}{2}}}$$ to be uniformly compact on a set of probability one, with set of limit points $$K_w = \{wf: f \in K\}$$. $K$ is the set set of absolutely continuous functions on $\lbrack 0, 1 \rbrack$ with $f(0) = 0 = f(1)$ and $$\int^1_0 \lbrack f'(t) \rbrack^2 dt \leqq 1.$$

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Barry R. James. "A Functional Law of the Iterated Logarithm for Weighted Empirical Distributions." Ann. Probab. 3 (5) 762 - 772, October, 1975. https://doi.org/10.1214/aop/1176996263

Information

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

zbMATH: 0347.60030
MathSciNet: MR402881
Digital Object Identifier: 10.1214/aop/1176996263

Subjects:
Primary: 60F15
Secondary: 60F20

Keywords: Law of the iterated logarithm , relatively compact sequence , Strassen-type limit set , weighted empirical distribution functions

Rights: Copyright © 1975 Institute of Mathematical Statistics

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Vol.3 • No. 5 • October, 1975
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