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arXiv · 2212.07804

Martingale Type, the Gamlen-Gaudet Construction and a Greedy Algorithm

Krystian Kazaniecki·Paul F. X. Müller
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Abstract

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}.

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Krystian Kazaniecki, Paul F. X. Müller. 2022-12-15. Martingale Type, the Gamlen-Gaudet Construction and a Greedy Algorithm. https://arxiv.org/abs/2212.07804

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