Spectral analysis of generalized weakly homogeneous random fields on nuclear spaces. (English. Ukrainian original) Zbl 0941.60011

Ponomarenko, A. I.

Theory Probab. Math. Stat. 53, 161-171 (1996); translation from Teor. Jmovirn. Mat. Stat. 53, 149-159 (1995).

The class of generalized random fields on locally compact groups is a natural extension of the class of generalized stationary stochastic processes and the class of weakly homogeneous random fields on the \(n\)-dimensional space \(R^n\). This paper deals with another extension of the classes. Random continuous linear functionals on the space of test measures on a locally convex space \(\mathcal Y\) are introduced to be generalized random fields on a locally convex space. Basic operations for generalized random fields on \(\mathcal Y\) are determined. Spectral representations of univariate and multivariate generalized random fields on nuclear space are obtained. Linear transformations of generalized homogeneous random fields on \(\mathcal Y\) are considered. The optimal linear filtering problem for random fields on \(\mathcal Y\) is investigated.

Reviewer: M.P.Moklyachuk (Kyïv)

MSC:

60B11	Probability theory on linear topological spaces
60G57	Random measures
28B05	Vector-valued set functions, measures and integrals
46F25	Distributions on infinite-dimensional spaces

Keywords:

random continuous functionals; nuclear spaces; spectral representation; correlation functional; filtering