A class of probabilistic measures in Hilbert space. (Russian) Zbl 0638.60008

Ryzhov, Yu. M.

Teor. Veroyatn. Mat. Stat., Kiev 37, 117-132 (1987).

Let H be a separable real Hilbert space and \(L_ 2(H)\) be the Hilbert space of Hilbert-Schmidt operators on H. A probabilistic measure \(\mu\) in H is called quadratic Gaussian (QG-measure) if with respect to this measure every continuous linear functional in H has the same distribution as some measurable centered quadratic functional with respect to some Gaussian measure in H. The characteristic functional of the QG-measure has the form \[ \phi_{{\mathcal K}}(z)=\exp \{-tr[i {\mathcal K}z+2^{-1} \ln (E-2i {\mathcal K}z)]\}, \] where \({\mathcal K}: H\to L_ 2(H)\) is a linear map. It is proved that \(\phi_{{\mathcal K}}(z)\) is a characteristic functional of some probabilistic measure in H if and only if the map \(\mathcal K\) has finite norm in \(L_ 2(H)\) or, equivalently, if the operator \({\mathcal K}^*{\mathcal K}: H\to H\) is nuclear. It is also shown that every QG- measure may be obtained as an extension of some canonical QG-measure on \(H_ 0\subset H\) corresponding to some quasi-nuclear imbedding \(H_ 0\to H\). This canonical measure is the image \(\nu\circ J\) of the cylindrical measure \(\nu\) in \(L_ 2(H)\) having characteristic functional \(\exp\{-tr[i Z+2^{-1} \ln (E-2i Z)]\}\), \(Z\in L_ 2(H)\), for some isomorphism \(J: L_ 2(H)\to H.\)
The weak closure of the class of the QG-measures is investigated. For a sequence \(\mu_ n\) of QG-measures, a necessary and sufficient condition of weak convergence is obtained. The limit measure has only the form of the composition \(\mu_ 0*\mu_ 1\) where \(\mu_ 0\) is a Gaussian measure and \(\mu_ 1\) is a QG-measure.

Reviewer: Yu.M.Ryzhov

MSC:

60B11	Probability theory on linear topological spaces
60B10	Convergence of probability measures

Keywords:

weak convergence; Hilbert-Schmidt operators; quadratic Gaussian; characteristic functional; quasi-nuclear imbedding; limit measure; Gaussian measure