## The Annals of Probability

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

### Extension du théorème de Cameron--Martin aux translations aléatoires

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 12 June 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1055425780

**Digital Object Identifier**

doi:10.1214/aop/1055425780

**Mathematical Reviews number (MathSciNet)**

MR1988473

**Zentralblatt MATH identifier**

1051.60039

**Subjects**

Primary: 60G15: Gaussian processes 60G30: Continuity and singularity of induced measures

Secondary: 28D05: Measure-preserving transformations

**Keywords**

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

#### Citation

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