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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.)
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