## Article

#### Title: A Characterization of p-uniformly Smooth Banach Spaces and Weak Laws of Large Numbers for d-dimensional Adapted Arrays

##### Issue: Volume 72 Series A Part 2 Year 2010
###### Abstract
In this paper, we introduce the concept of multiparameter martingale differences and provide a characterization of $p$-uniformly smooth Banach spaces in terms of an inequality for multiparameter martingale differences. Then we apply this result to establish some weak laws of large numbers, where the classical degenerate convergence criterion and Kolmogorov-Feller weak law of large numbers will be extended to $d$-dimensional adapted arrays in Banach spaces. Some special cases of our results are presented as corollaries, and illustrative examples are provided.

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