STRASSEN, V., Almost sure behavior of sums of independent random variables and martingales, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. II: Contributions to
In this paper atomic decompositions for two-parameter vector-valued martingales are given. With the help of the atomic decompositions
the relations between the mutual embedding of two-parameter vector-valued martingale spaces and geometric properties of Banach
spaces are investigated. Our study shows that geometric properties of Banach spaces determine the embedding of martingale
spaces and conversely the latter can characterize the former.
Several interpolation theorems on martingale Hardy spaces over weighted measure spaces are given. Our proofs are based on
the atomic decomposition of martingale Hardy spaces over weighted measure spaces. As applications of interpolation theorems,
some inequalities of martingale transform operator are obtained.
In a one-parameter model for evolution of random trees strong law of large numbers and central limit theorem are proved for the number of vertices with low degree. The proof is based on elementary martingale theory.
Some atomic decomposition theorems are proved in vector-valued weak martingale Hardy spaces wpΣα(X), wpQα(X) and wDα(X). As applications of atomic decompositions, a sufficient condition for sublinear operators defined on some vector-valued
weak martingale Hardy spaces to be bounded is given. In particular, some weak versions of martingale inequalities for the
operators f*, S(p)(f) and σ(p)(f) are obtained.