The Annals of Probability

Extension du théorème de Cameron--Martin aux translations aléatoires

Xavier Fernique

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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 $\gamma$.

Article information

Ann. Probab., Volume 31, Number 3 (2003), 1296-1304.

First available in Project Euclid: 12 June 2003

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes 60G30: Continuity and singularity of induced measures
Secondary: 28D05: Measure-preserving transformations

Théorème de Cameron--Martin probabilité gaussienne absolue continuité Cameron--Martin theorem Gaussian probability absolute continuity


Fernique, Xavier. Extension du théorème de Cameron--Martin aux translations aléatoires. Ann. Probab. 31 (2003), no. 3, 1296--1304. doi:10.1214/aop/1055425780.

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