Gikhman-Skorokhod spectral conditions of equivalence of Gaussian measures corresponding to homogeneous random fields (Q2737025)

Let \(P_{1}\) and \(P_{2}\) be two probability measures on a measurable space \((\Omega, F)\) and let a random function \(\xi\) be a homogeneous random field on Euclidean space \(R^{N}\) with respect to these two measures. Let \(T\) be a subset of \(R^{N}\) and let \(P_{1}^{T}\), \(P_{2}^{T}\) be restrictions of the measures \(P_{1}\) and \(P_{2}\), respectively, to the \(\sigma\)-algebra generated by values of \(\xi\) on \(T.\) It is supposed that measures \(P_{1}\) and \(P_{2}\) have spectral densities \(f_{1}\) and \(f_{2},\) respectively. It is known that measures \(P_{1}^{T}\) and \(P_{2}^{T}\) may be only either equivalence or orthogonal. The author is interested in the conditions for equivalence or orthogonality of the considered measures which are described in terms of spectral densities of the random field \(\xi.\) The main stuff of the paper is concerned on sufficient conditions of equivalence of the above-mentioned measures.