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October, 1975 A Maximal Inequality and Dependent Strong Laws
D. L. McLeish
Ann. Probab. 3(5): 829-839 (October, 1975). DOI: 10.1214/aop/1176996269

Abstract

This paper contains a general dependent extension of Doob's inequality for martingales, $E(\max_{i\leqq n} S_i^2) \leqq 4ES_n^2$. This inequality is then used to extend the martingale convergence theorem for $L_2$ bounded variables, and to prove strong laws under dependent assumptions. Strong and $\varphi$-mixing variables are shown to satisfy the conditions of these theorems and hence strong laws are proved as well for these.

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D. L. McLeish. "A Maximal Inequality and Dependent Strong Laws." Ann. Probab. 3 (5) 829 - 839, October, 1975. https://doi.org/10.1214/aop/1176996269

Information

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

zbMATH: 0353.60035
MathSciNet: MR400382
Digital Object Identifier: 10.1214/aop/1176996269

Subjects:
Primary: 60F15
Secondary: 60G45

Keywords: dependent variables , Doob's inequality , martingale convergence theorem , Mixing , strong law

Rights: Copyright © 1975 Institute of Mathematical Statistics

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Vol.3 • No. 5 • October, 1975
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