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July 2003 Extension du théorème de Cameron--Martin aux translations aléatoires
Xavier Fernique
Ann. Probab. 31(3): 1296-1304 (July 2003). DOI: 10.1214/aop/1055425780


Let G be a Gaussian vector taking its values in a separable Fréchet space E. We denote by $\gamma$ its law and by $(H,\Vert\!\cdot\!\Vert)$ its reproducing Hilbert space. Moreover, let X be an E-valued random vector of law $\mu$. In the first section, we prove that if $\mu$ is absolutely continuous relative to $\gamma$, then there exist necessarily a Gaussian vector $G'$ of law $\gamma$ and an H-valued random vector Z such that $G' + Z$ has the law $\mu$ of X. This fact is a direct consequence of concentration properties of Gaussian vectors and, in some sense, it is an unexpected achievement of a part of the Cameron--Martin theorem.

In the second section, using the classical Cameron--Martin theorem and rotation invariance properties of Gaussian probabilities, we show that, in many situations, such a condition is sufficient for $\mu$ being absolutely continuous relative to $\gamma$.


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Xavier Fernique. "Extension du théorème de Cameron--Martin aux translations aléatoires." Ann. Probab. 31 (3) 1296 - 1304, July 2003.


Published: July 2003
First available in Project Euclid: 12 June 2003

zbMATH: 1051.60039
MathSciNet: MR1988473
Digital Object Identifier: 10.1214/aop/1055425780

Primary: 60G15 , 60G30
Secondary: 28D05

Keywords: absolue continuité , Absolute continuity , Cameron--Martin theorem , Gaussian probability , probabilité gaussienne , Théorème de Cameron--Martin

Rights: Copyright © 2003 Institute of Mathematical Statistics

Vol.31 • No. 3 • July 2003
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