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April, 1980 A Characterization of Vitali Conditions in Terms of Maximal Inequalities
Annie Millet, Louis Sucheston
Ann. Probab. 8(2): 339-349 (April, 1980). DOI: 10.1214/aop/1176994781

Abstract

Vitali conditions $V, V', V_p, 1 \leqslant p < \infty$, on $\sigma$-algebras indexed by a directed set, are shown to hold if and only if the maximal inequality \begin{equation*}\tag{1} P(\text{essential} \lim \sup X_t \geqslant \alpha) \leqslant K \lim \sup_{T^\ast}E(X_\tau)/\alpha\end{equation*} holds for all adapted positive processes $(X_t)$, and all positive numbers $\alpha$. Here $K$ is a constant which may be taken equal to 1, and $T^\ast$ is the appropriate directed set of stopping times: for $V, T^\ast$ is the set of simple stopping times; for $V', T^\ast$ is the set of simple ordered stopping times; for $V_p, T^\ast$ is the set of multivalued stopping times with overlap going to zero in $L_p$. The inequality (1) is true whatever be the $\sigma$-algebras, provided that essential $\lim \sup$ is replaced by stochastic $\lim \sup$.

Citation

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Annie Millet. Louis Sucheston. "A Characterization of Vitali Conditions in Terms of Maximal Inequalities." Ann. Probab. 8 (2) 339 - 349, April, 1980. https://doi.org/10.1214/aop/1176994781

Information

Published: April, 1980
First available in Project Euclid: 19 April 2007

zbMATH: 0428.60053
MathSciNet: MR566598
Digital Object Identifier: 10.1214/aop/1176994781

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

Rights: Copyright © 1980 Institute of Mathematical Statistics

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Vol.8 • No. 2 • April, 1980
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