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Concrete Representation of Martingales

1998
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Electronic Journal of Probability
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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 1 0 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 ( n k=1 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

doi:10.1214/ejp.v3-37
fatcat:fjyfcoak2bge7p3nfuyvakdr6m