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April, 1973 Maximal Inequalities and the Law of the Iterated Logarithm
William F. Stout
Ann. Probab. 1(2): 322-328 (April, 1973). DOI: 10.1214/aop/1176996985


A supermartingale maximal inequality is derived. A maximal inequality is derived for arbitrary random variables $\{S_n, n \geqq 1\}$ (let $S_0 = 0$) satisfying $E\exp\lbrack u(S_{m + n} - S_m) \rbrack \leqq \exp(Knu^2)$ for all real $u$, all integers $m \geqq 0$ and $n \geqq 1$, and some constant $K$. These two maximal inequalities are used to derive upper half laws of the iterated logarithm for supermartingales, multiplicative random variables, and random variables not satisfying particular dependence assumptions.


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William F. Stout. "Maximal Inequalities and the Law of the Iterated Logarithm." Ann. Probab. 1 (2) 322 - 328, April, 1973.


Published: April, 1973
First available in Project Euclid: 19 April 2007

zbMATH: 0262.60016
MathSciNet: MR353428
Digital Object Identifier: 10.1214/aop/1176996985

Primary: 60F15
Secondary: 60G40 , 60G45 , 60G99

Keywords: equinormed strongly multiplicative random variables , generalized Gaussian random variables , Law of the iterated logarithm , martingale , maximal inequality , multiplicative random variables , supermartingale

Rights: Copyright © 1973 Institute of Mathematical Statistics

Vol.1 • No. 2 • April, 1973
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