A sufficient condition for the equivalence of probability measures corresponding to Gaussian homogeneous random fields. (English. Ukrainian original) Zbl 0998.60036

Krasnits’kyj, S. M.

Theory Probab. Math. Stat. 64, 85-91 (2002); translation from Teor. Jmovirn. Mat. Stat. 64, 75-81 (2001).

The author presents a sufficient condition of equivalence of measures induced by generalized (as well as usual) homogeneous Gaussian centered random fields. This condition is formulated in terms of the generalized functions \(K\) such that the correlation function of the field \(\xi(\cdot)\) can be represented as \(E\xi(\varphi)\xi(\psi)=K(\varphi\ast\psi)\), where \(\varphi\ast\psi\) is a convolution of the basic functions \(\varphi\) and \(\psi\). The integral representation of multi-parameter functions, properties of the Sobolev functional spaces and embedding theorems play an important role in analysis and reformulation of the equivalence condition.

Reviewer: N.M.Zinchenko (Kyïv)

MSC:

60G30

Continuity and singularity of induced measures

Keywords:

Gaussian field; generalized field; equivalence of measures; correlation function