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December, 1978 A Uniform Law of the Iterated Logarithm for Classes of Functions
R. Kaufman, Walter Philipp
Ann. Probab. 6(6): 930-952 (December, 1978). DOI: 10.1214/aop/1176995385


Let $\{\xi_k, k \geqslant 1\}$ be a sequence of random variables uniformly distributed over $\lbrack 0, 1\rbrack$ and let $\mathscr{F}$ be a class of functions on $\lbrack 0, 1\rbrack$ with $\int^1_0 f(x) dx = 0$. In this paper we give upper and lower bounds for $\sup_{f \in \mathscr{F}}|\sigma_{k \leqslant N}f(\xi_k)|$ for the class of functions of variation bounded by 1 and for the class of functions satisfying a Lipschitz condition.


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R. Kaufman. Walter Philipp. "A Uniform Law of the Iterated Logarithm for Classes of Functions." Ann. Probab. 6 (6) 930 - 952, December, 1978.


Published: December, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0404.60041
MathSciNet: MR512412
Digital Object Identifier: 10.1214/aop/1176995385

Primary: 60F15
Secondary: 10K15 , 42A44

Keywords: Hilbert space valued random variables , lacunary sequences , Law of the iterated logarithm , mixing sequences of random variables

Rights: Copyright © 1978 Institute of Mathematical Statistics

Vol.6 • No. 6 • December, 1978
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