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Concrete representation of martingales. (English) Zbl 0916.60044

Let \((d_n)\) be a Bochner integrable martingale difference sequence taking values in a Banach space \(X\). A sequence of Bochner measurable functions \((e_n:[0,1]^n\to X)\) is constructed such that \((d_n)\) has the same probability law as \((e_n)\) and such that for almost every \(x_1,\dots,x_{n-1}\) with respect to the Lebesgue measure \(\int^1_0 e_n(x_1, \dots,x_n)dx_n=0\). Similar representation theorems are proved for tangent sequences and sequences satisfying condition (C.I.), introduced by S. Kwapień and W. A. Woyczyński. A version of a Skorokhod like imbedding theorem for a Bochner integrable martingale difference sequence taking values in a Banach space \(X\) is also proved.

MSC:

60G42 Martingales with discrete parameter
60H99 Stochastic analysis
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