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Concrete representation of martingales

Stephen J. Montgomery-Smith
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Abstract

Let (f_n) be a mean zero vector valued martingale sequence. Then there exist vector valued functions (d_n) from [0,1]^n such that int_0^1 d_n(x_1,...,x_n) dx_n = 0 for almost all x_1,...,x_{n-1}, and such that the law of (f_n) is the same as the law of (sum_{k=1}^n d_k(x_1,...,x_k)) . 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

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Stephen J. Montgomery-Smith. 1999-12-03. Concrete representation of martingales. https://arxiv.org/abs/math/9806022

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