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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Derivation, $ L\sp \Psi$-bounded martingales and covering conditions


Author: Michel Talagrand
Journal: Trans. Amer. Math. Soc. 293 (1986), 257-291
MSC: Primary 28A15; Secondary 46G05, 60G42
MathSciNet review: 814922
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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

DOI: http://dx.doi.org/10.1090/S0002-9947-1986-0814922-8
PII: S 0002-9947(1986)0814922-8
Keywords: Derivation, martingales, covering conditions, Orlicz spaces
Article copyright: © Copyright 1986 American Mathematical Society