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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Derivation, $L^ \Psi$-bounded martingales and covering conditions
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by Michel Talagrand PDF
Trans. Amer. Math. Soc. 293 (1986), 257-291 Request permission

Abstract:

Let $(\Omega , \Sigma , P)$ be a complete probability space. Let ${({\Sigma _t})_{t \in J}}$ be a directed family of sub-$\sigma$-algebras of $\Sigma$. Let $(\Phi , \Psi )$ be a pair of conjugate Young functions. We investigate the covering conditions that are equivalent to the essential convergence of ${L^\Psi }$-bounded martingales. We do not assume that either $\Phi$ or $\Psi$ satisfy the ${\Delta _2}$ condition. We show that when $\Phi$ satisfies condition Exp, that is when there exists an $a > 0$ such that $\Phi (u) \leq \operatorname {exp} au$ for each $u \ge 0$, the essential convergence of ${L^\Psi }$-bounded martingales is equivalent to the classical covering condition ${V_\Phi }$. This covers in particular the classical case $\Psi (t) = t{(\operatorname {log} t)^ + }$. The growth condition Exp on $\Phi$ cannot be relaxed. When $J$ contains a countable cofinite set, we show that the essential convergence of ${L^\Psi }$-bounded martingales is equivalent to a covering condition ${D_\Phi }$ (that is weaker than ${V_\Phi }$). When $\Phi$ fails condition Exp, condition ${D_\Phi }$ is optimal. Roughly speaking, in the case of ${L^1 }$-bounded martingales, condition ${D_\Phi }$ means that, locally, the Vitali condition with finite overlap holds. We also investigate the case where $J$ does not contain a countable cofinal set and $\Phi$ fails condition Exp. In this case, it seems impossible to characterize the essential convergence of ${L^\Psi }$-bounded martingales by a covering condition. Using the Continuum Hypothesis, we also produce an example where all equi-integrable ${L^1 }$-bounded martingales, but not all ${L^1 }$-bounded martingales, converge essentially. Similar results are also established in the derivation setting.
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Additional Information
  • © Copyright 1986 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 293 (1986), 257-291
  • MSC: Primary 28A15; Secondary 46G05, 60G42
  • DOI: https://doi.org/10.1090/S0002-9947-1986-0814922-8
  • MathSciNet review: 814922