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Optimum bounds for the distributions of martingales in Banach spaces. (English) Zbl 0836.60015
Summary: 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.

MSC:
60E15 Inequalities; stochastic orderings
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60G42 Martingales with discrete parameter
60G50 Sums of independent random variables; random walks
60F10 Large deviations
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