## Extension of the Cameron-Martin theorem to random translations. (Extension du théorème de Cameron-Martin aux translations aléatoires.)(French)Zbl 1051.60039

Let $$E$$ denote a separable Fréchet space with Gaussian probability $$\gamma$$ and corresponding reproducing-kernel Hilbert space $$H$$. Theorem 1 generalizes the (classical) Cameron-Martin theorem to random translations taking values in $$H$$, characterizing under which conditions a distribution $$\mu = D\cdot \gamma$$, i.e. absolutely continuous w.r.t. $$\gamma$$, is representable as the distribution of a random variable $$G' + Z$$, where $$G'$$ has distribution $$\gamma$$ and $$Z$$ is $$H$$-valued (“random translation”). [For the background see e.g. the author, “Fonctions aléatoires gaussiens, vecteurs aléatoires gaussiens” (Les Publications C.R.M. Montréal, 1977)].
The second problem is concerned with a converse problem, to find conditions under which the law of $$G + Z$$ is absolutely continuous w.r.t. the law $$\gamma$$ of $$G$$. The proof relies on properties of the concentration of Gaussian vectors [see e.g. M. Ledoux and M. Talegrand, “Probability in Banach spaces. Isoperimetry and processes” (1991; Zbl 0748.60004)] and the Kantorovich-Rubinstein theorem [see e.g. S. T. Rachev and L. Rüschendorf, “Mass transport problems”. Vol. 1 (1998; Zbl 0990.60500)].

### MSC:

 60G15 Gaussian processes 60G30 Continuity and singularity of induced measures 28D05 Measure-preserving transformations 28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)

### Citations:

Zbl 0748.60004; Zbl 0990.60500
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