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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.
##### MSC:
 60F15 Strong limit theorems 60G42 Martingales with discrete parameter