Two-Parameter Banach Space Valued Strong Martingales and Characterizations of p-Smoothable Spaces

In this paper we introduce the concept of two-parameterB-valued strong martingales and investigate some features of these strong martingales. We also characterizep-smoothable Banach spaces in terms of these strong martingales.

Davis's Inequalities for Two-Parameter Banach Space Valued Strong Martingales and Geometrical Properties of Banach Spaces

In this paper we investigated theL 1 norm inequalities of theP square and the maximal functions of two-parameterB-valued strong martingales, which can be applied to characterizep-smoothness andq-convexity of Banach spaces.

The Hjek-Rnyi Inequlity For Banach Space Valued Random Variable Sequences and Its Application

In this paper we prove the following Hajek Renyi inequality:Let 0<p≤1 ,then for any Banach space B , any L p integrable B valued random variable sequence {D n,n≥1} ,any real number sequence {b n,n≥1} with 0<b n↑∞ ,any integer n≥1 ,there exits a constant C=C p>0 (only depending on p ) such thatP( sup j≥nji=1D ib j≥ε)≤Cε -p (∞j=n+1E‖D j‖ pb p j+nj=1E‖D j‖ pb p n) In the other direction,we prove some strong laws of large numbers and the integrability of the maximal functions for B valued random variable sequences by using this inequality and the Hajeck Renyi inequality we have obtained recently.Some known results are extended and improved.