Asymptotic properties of finite dimensional conditional distributions of spherically symmetric measures on a locally convex space.

*(English, Russian)*Zbl 1158.28302
Russ. Math. 49, No. 3, 67-74 (2005); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2005, No. 3, 71-78 (2005).

From the introduction: We consider finite dimensional projections of spherically symmetric measures on a locally convex space. We construct these projections as combined functions of distributions of finite systems of measurable linear functionals belonging to an orthonormal basis of the reproducing Hilbert space of the Gaussian measure which generates the given spherically symmetric measure. Then, for any projection (a finite system of basis functionals) we consider the corresponding conditional distribution of a fixed subsystem of functionals with respect to the other functionals of the same system. We prove that any such distribution almost sure converges to the Gaussian distribution as the dimension tends to infinity. Also we establish relationship of the obtained results with the logarithmic derivatives of spherically symmetric measures.

##### MSC:

28C20 | Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) |