A description of norm-convergent martingales on vector-valued LP-spaces

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:323

Article

Cullender, Stuart F. ; Labuschagne, Coenraad C. A.

关键词: filtration;    convergent martingale;    Banach lattice;    Banach space;   

DOI  :  10.1016/j.jmaa.2005.10.032

来源: Elsevier

【 摘 要 】

Norm-convergent martingales on tensor products of Banach spaces are considered in a measure-free setting. As a consequence, we obtain the following characterization for convergent martingales on vector-valued L-p-spaces: Let (Omega, Sigma, mu) be a probability space, X a Banach space and (Sigma(n)) an increasing sequence of sub sigma-algebras of Sigma. In order for (f(n), Sigma(n))(n=1)(infinity) to be a convergent martingale in L-p(mu, X) (1 <= p < infinity) it is necessary and sufficient that, for each i epsilon N, there exists a convergent martingale (x(i)((n)), Sigma(n))(n=1)(infinity) in L-p(mu) and y(i) epsilon X such that, for each n epsilon N, we have [GRAPHICS] where parallel to Sigma(infinity)(i=1)vertical bar lim(n ->infinity)x(i)((n))vertical bar parallel to L-p(mu) < infinity and lim(i ->infinity)parallel to gamma(i)parallel to -> 0. (c) 2005 Elsevier Inc. All rights reserved.

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