Iosif Pinelis
Published 2012-08-10, updated 2012-12-09Version 2
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 the 2-smooth Banach spaces. In turn, it leads to best-order bounds on moments of the sums of independent random vectors in any separable Banach spaces. Although the emphasis is put on the infinite-dimensional martingales, most of the results seem to be new even for the one-dimensional ones. Moreover, the bounds on the Rosenthal-Burkholder type of moments seem to be to certain extent new even for the sums of independent real-valued random variables. Analogous inequalities for (one-dimensional) supermartingales are given.
Comments: This is an original draft of the published paper under the same title. The primary reason for posting this draft is to provide for references to the Appendix there, which was not included into the published paper. The previous arXiv version (Version 1) of this posting contained a number of typos. I have since found a later and better draft, which is represented by this latter posting
Journal: Ann. Probab. 22 (1994), no. 4, 1679--1706; (see also Correction) Ann. Probab. 27 (1999), no. 4, 2119
Hanson-Wright inequality in Banach spaces
Maximal inequalities for centered norms of sums of independent random vectors
Some Classes Of Distributions On The Non-Negative Lattice
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