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

Author(s): Nguyen Van Quang and Nguyen Van Huan
Issue: Volume 72 Series A Part 2 Year 2010
Pages: 344 -- 358
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.
AMS (2000) subject classification. Primary 60F05, 60G50, 60G42.
Keywords and phrases: $p$-uniformly smooth Banach space, adapted array, martingale difference array, weak law of large numbers.