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.References
- Kenneth A. Astbury, The order convergence of martingales indexed by directed sets, Trans. Amer. Math. Soc. 265 (1981), no. 2, 495–510. MR 610961, DOI 10.1090/S0002-9947-1981-0610961-6
- Miguel de Guzmán, Differentiation of integrals in $R^{n}$, Lecture Notes in Mathematics, Vol. 481, Springer-Verlag, Berlin-New York, 1975. With appendices by Antonio Córdoba, and Robert Fefferman, and two by Roberto Moriyón. MR 0457661, DOI 10.1007/BFb0081986 C. A. Hayes and C. Y. Pauc, Derivation and martingales, Springer-Verlag, 1970.
- C. A. Hayes, Necessary and sufficient conditions for the derivation of integrals of $L_{\psi }$-functions, Trans. Amer. Math. Soc. 223 (1976), 385–395. MR 427554, DOI 10.1090/S0002-9947-1976-0427554-4
- K. Krickeberg, Convergence of martingales with a directed index set, Trans. Amer. Math. Soc. 83 (1956), 313–337. MR 91328, DOI 10.1090/S0002-9947-1956-0091328-4
- Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. II, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR 540367, DOI 10.1007/978-3-662-35347-9
- Annie Millet, Sur la caractérisation des conditions de Vitali par la convergence essentielle des martingales, C. R. Acad. Sci. Paris Sér. A-B 287 (1978), no. 13, A887–A890 (French, with English summary). MR 551772
- Annie Millet and Louis Sucheston, On covering conditions and convergence, Measure theory, Oberwolfach 1979 (Proc. Conf., Oberwolfach, 1979) Lecture Notes in Math., vol. 794, Springer, Berlin, 1980, pp. 431–454. MR 577989
- Annie Millet and Louis Sucheston, A characterization of Vitali conditions in terms of maximal inequalities, Ann. Probab. 8 (1980), no. 2, 339–349. MR 566598
- Annie Millet and Louis Sucheston, On convergence of $L_{1}$-bounded martingales indexed by directed sets, Probab. Math. Statist. 1 (1980), no. 2, 151–169 (1981). MR 626308
- Kôsaku Yosida and Edwin Hewitt, Finitely additive measures, Trans. Amer. Math. Soc. 72 (1952), 46–66. MR 45194, DOI 10.1090/S0002-9947-1952-0045194-X A. C. Zaanen, Linear analysis, North-Holland, Amsterdam, 1953.
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