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December, 1981 A Geometrical Characterization of Banach Spaces in Which Martingale Difference Sequences are Unconditional
D. L. Burkholder
Ann. Probab. 9(6): 997-1011 (December, 1981). DOI: 10.1214/aop/1176994270

Abstract

We study Banach-space-valued martingale transforms and, in particular, characterize those Banach spaces for which the classical theorems of the real-valued case carry over. For example, if $B$ is a Banach space and $1 < p < \infty$, then there exists a positive real number $c_p$ such that $\|\epsilon_1d_1 + \cdots + \epsilon_n d_n \|_p \leq c_p \|d_1 + \cdots + d_n \|_p$ for all $B$-valued martingale difference sequences $d = (d_1, d_2,\cdots)$ and all numbers $\epsilon_1, \epsilon_2,\cdots$ in $\{-1, 1\}$ if and only if there is a symmetric biconvex function $\zeta$ on $B \times B$ satisfying $\zeta(0, 0) > 0$ and $\zeta(x, y) \leq | x + y | \text{if} | x | \leq 1 \leq | y |$.

Citation

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D. L. Burkholder. "A Geometrical Characterization of Banach Spaces in Which Martingale Difference Sequences are Unconditional." Ann. Probab. 9 (6) 997 - 1011, December, 1981. https://doi.org/10.1214/aop/1176994270

Information

Published: December, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0474.60036
MathSciNet: MR632972
Digital Object Identifier: 10.1214/aop/1176994270

Subjects:
Primary: 60G42
Secondary: 46B20, 46C05, 60G46

Rights: Copyright © 1981 Institute of Mathematical Statistics

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Vol.9 • No. 6 • December, 1981
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