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1998 Concrete Representation of Martingales
Stephen Montgomery-Smith
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Electron. J. Probab. 3: 1-15 (1998). DOI: 10.1214/EJP.v3-37


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 $\int_0^1 d_n(x_1,\dots,x_n)\,dx_n = 0$ for almost all $x_1,\dots,x_{n-1}$, and such that the law of $(f_n)$ is the same as the law of $(\sum_{k=1}^n d_k(x_1,\dots,x_k))$. 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.


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Stephen Montgomery-Smith. "Concrete Representation of Martingales." Electron. J. Probab. 3 1 - 15, 1998.


Accepted: 2 December 1998; Published: 1998
First available in Project Euclid: 29 January 2016

zbMATH: 0916.60044
MathSciNet: MR1658686
Digital Object Identifier: 10.1214/EJP.v3-37

Primary: 60G42
Secondary: 60H05

Keywords: concrete representation , condition (C.I.) , martingale , Skorohod representation , tangent sequence , UMD

Vol.3 • 1998
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