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 Teor. Veroyatnost. i Primenen., 1998, Volume 43, Issue 3, Pages 588–598 (Mi tvp1563)

Short Communications

Atomic decompositions and inequalities for vector-valued discrete-time martingales

F. Weisza, Yu. S. Mishurab

a Department of Numerical Analysis, Eötvös University, Hungary
b National Taras Shevchenko University of Kyiv, Faculty of Mechanics and Mathematics

Abstract: We consider martingales with discrete time taking values in a Banach lattice $X$ that has UMD-property (UMD means unconditionality of martingale differences). We suppose that the UMD-lattice $X$ consists of real-valued functions. Two notions of maximal value for such martingales are introduced (in the case of real-valued martingales these notions are the same and also coincide with the notion of usual maximal value). We also introduce the notion of quadratic variation and both usual and predictable classes of martingale spaces corresponding to maximal values and quadratic variation. The equivalence of these classes is established. In particular, Davis inequalities are proved with the help of atomic decompositions. The case of a regular stochastic basis is considered separately.

Keywords: vector-valued martingales with discrete time, UMD-lattice, maximal value, quadratic variation, Burkholder–Davis–Gundy inequalities, atomic decomposition, regular stochastic basis.

DOI: https://doi.org/10.4213/tvp1563

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English version:
Theory of Probability and its Applications, 1999, 43:3, 487–496

Bibliographic databases:

Citation: F. Weisz, Yu. S. Mishura, “Atomic decompositions and inequalities for vector-valued discrete-time martingales”, Teor. Veroyatnost. i Primenen., 43:3 (1998), 588–598; Theory Probab. Appl., 43:3 (1999), 487–496

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tvp1563
• https://doi.org/10.4213/tvp1563
• http://mi.mathnet.ru/eng/tvp/v43/i3/p588

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This publication is cited in the following articles:
1. Cheng R.Y., Gan S.X., “A new characterization of p–smoothable spaces”, Acta Mathematica Hungarica, 87:1–2 (2000), 121–133
2. Riyan C., Shixin G., “Atomic decompositions for two-parameter vector-valued martingales and two-parameter vector-valued martingale spaces”, Acta Math Hungar, 93:1–2 (2001), 7–25
3. Weisz F., “Hardy spaces and convergence of vector–valued Vilenkin–Fourier series”, Publicationes Mathematicae–Debrecen, 71:3–4 (2007), 413–424
4. Weisz F., “Almost everywhere convergence of Banach space–valued Vilenkin–Fourier series”, Acta Mathematica Hungarica, 116:1–2 (2007), 47–59
5. Hou Y.-L., Ren Y.-B., “Vector–valued weak martingale Hardy spaces and atomic decompositions”, Acta Mathematica Hungarica, 115:3 (2007), 235–246
6. Zhang X., Zhang Ch., “Atomic decompositions of Banach lattice-valued martingales”, Statistics & Probability Letters, 82:3 (2012), 664–671
7. Yang A., “Bounded Operators on Vector-Valued Weak Orlicz Martingale Spaces”, Acta Math. Hung., 152:1 (2017), 186–200
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