zbMATH — the first resource for mathematics

On a spectral condition for the equivalence of Gaussian measures corresponding to random fields. (English. Ukrainian original) Zbl 0956.60010
Theory Probab. Math. Stat. 60, 95-104 (2000); translation from Teor. Jmovirn. Mat. Stat. 60, 85-94 (1999).
The author considers two generalized Gaussian homogeneous random fields in \(R^{N}.\) Let \(P_1^{\Phi}\) and \(P_2^{\Phi}\) be measures generated by these fields. Let \(\Phi=\Phi(T)\) be an open bounded set in \(R^{N}.\) Let \(f_1\) and \(f_2\) be spectral densities corresponding to measures \(P_1^{\Phi}\) and \(P_2^{\Phi}.\) New conditions of equivalence of measures \(P_1^{\Phi}\) and \(P_2^{\Phi}\) are found. The main result is the following theorem: Let the spectral density \(f_1(\lambda)\) satisfy the condition \(f_1(\lambda)>\) const\(\cdot|{\widetilde k}_0(\lambda)|^2,\) where \(k_0(\lambda)\) is a generalized function with nonempty compact support and \({\widetilde k}_0(\lambda)\) is the Fourier transform of \(k_0(\lambda).\) If the function \((f_1 -f_2) (\lambda)\cdot |{\widetilde k}_0(\lambda)|^{-2}\) belongs to the space \(L_{p}(R^{N})\) for some \(p\in [1,2],\) then the measures \(P_1^{\Phi}\) and \(P_2^{\Phi}\) are equivalent.
60F15 Strong limit theorems
60G42 Martingales with discrete parameter