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] Weisz , F. 1994 Martingale Hardy Spaces and their Applications in Fourier Analysis Lecture Notes in Math. 1568. [7] Long , R

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operator . Studia Math ., 110 : 149 – 167 , 1994 . [3] W . Chen and P.D . Liu . Several weak-type weighted inequalities in Orlicz martingale classes . Acta Mathematica Scientia , 31 : 1041 – 1050 , 2011 . [4] W . Chen and P.D . Liu

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Abstract  

Characterizations of the two-dimensional H 1, BMO and VMO martingale spaces generated by bounded Vilenkin systems via conjugate martingale transforms are studied.

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Abstract  

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.

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For discrete martingales we show that empirical measures related to the central limit theorem when appropriately weighted converge weakly toward a Gaussian measure,for almost all trajectories.Thi result enables u to derive a weighted trong law of the large number for which we pecify both weak and trong rate of convergence.

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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.

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Reconstruction theorems for martingales with respect to regular filtration are proved provided that the majorant of the martingale satisfies some specified condition. The ob-tained results are applied to obtain formulas for restoration of coeffcients for multiple Haar series.

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Abstract  

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.

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Abstract  

Some atomic decomposition theorems are proved in vector-valued weak martingale Hardy spaces w p Σα(X), w p Q α(X) and wD α(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.

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The aim of this paper is to represent any continuous local martingale as an almost sure limit of a nested sequence of simple, symmetric random walk, time changed by a discrete quadratic variation process. One basis of this is a similar construction of Brownian motion. The other major tool is a representation of continuous local martingales given by Dambis, Dubins and Schwarz (DDS) in terms of Brownian motion time-changed by the quadratic variation. Rates of convergence (which are conjectured to be nearly optimal in the given setting) are also supplied. A necessary and sufficient condition for the independence of the random walks and the discrete time changes or equivalently, for the independence of the DDS Brownian motion and the quadratic variation is proved to be the symmetry of increments of the martingale given the past, which is a reformulation of an earlier result by Ocone [8].

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