Concrete representation of martingales

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Let (fₙ) be a mean zero vector valued martingale sequence. Then there exist vector valued functions (dₙ) from [0,1]ⁿ such that int₀¹ dₙ(x₁,...,xₙ) dxₙ = 0 for almost all x₁,...,xₙ₋₁, and such that the law of (fₙ) is the same as the law of (sumₖ₌₁ⁿ dₖ(x₁,...,xₖ)). Similar results for tangent sequences and sequences satisfying condition (C.I.) are presented. We also present a weaker version of a result of McConnell that provides a Skorohod like representation for vector valued martingales. This paper may be found at http://math.missouri.edu/ stephen/preprints