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October, 1994 Optimum Bounds for the Distributions of Martingales in Banach Spaces
Iosif Pinelis
Ann. Probab. 22(4): 1679-1706 (October, 1994). DOI: 10.1214/aop/1176988477


A general device is proposed, which provides for extension of exponential inequalities for sums of independent real-valued random variables to those for martingales in the 2-smooth Banach spaces. This is used to obtain optimum bounds of the Rosenthal-Burkholder and Chung types on moments of the martingales in 2-smooth Banach spaces. In turn, it leads to best-order bounds on moments of sums of independent random vectors in any separable Banach spaces. Although the emphasis is put on infinite-dimensional martingales, most of the results seem to be new even for one-dimensional martingales. Moreover, the bounds on moments of the Rosenthal-Burkholder type seem to be to a certain extent new even for sums of independent real-valued random variables. Analogous inequalities for (one-dimensional) supermartingales are given.


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Iosif Pinelis. "Optimum Bounds for the Distributions of Martingales in Banach Spaces." Ann. Probab. 22 (4) 1679 - 1706, October, 1994.


Published: October, 1994
First available in Project Euclid: 19 April 2007

zbMATH: 0836.60015
MathSciNet: MR1331198
Digital Object Identifier: 10.1214/aop/1176988477

Primary: 60E15
Secondary: 60B12 , 60F10 , 60G42 , 60G50

Keywords: 2-smooth Banach spaces , bounds on moments , Distribution inequalities , Exponential inequalities , martingales in Banach spaces , Sums of independent random variables

Rights: Copyright © 1994 Institute of Mathematical Statistics


Vol.22 • No. 4 • October, 1994
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