Martingale Type, the Gamlen-Gaudet Construction and a Greedy Algorithm
| Autor: | Kazaniecki, Krystian, Müller, Paul F. X. |
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| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | In the present paper we identify those filtered probability spaces $(\Omega,\, \mathcal{F},\, \left(\mathcal{F}_n\right),\, \mathbb{P})$ that determine already the martingale type of a Banach space $X$. We isolate intrinsic conditions on the filtration $(\mathcal{F}_n)$ of purely atomic $\sigma$-algebras which determine that the upper $\ell^p$ estimates \[ \|f\|_{L^p(\Omega,\, X)}^p\leq C^p\left( \|\mathbb{E} f|\mathcal{F}_0\|^p_{L^p(\Omega,\, X)}+\sum_{n=1}^{\infty} \|\Delta_n f\|^p_{L^p(\Omega,\, X)}\right),\qquad f\in L^p(\Omega,X)\] imply that the Banach space $X$ is of martingale type $p$. Our paper complements \mbox{G. Pisier's} investigation \cite{Pisier1975} and continues the work by S. Geiss and second named author in \cite{Geiss2008}. Comment: We added: References, additional explanations, improvements. We remove typos and misprints |
| Databáze: | arXiv |
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