Local Asymptotical Normality for Stationary Gaussian Sequences with Degenerate Spectral Density

The property of local asymptotical normality (at a point $\theta_0$) is proved for a stationary Gaussian sequence with spectral density $f(\lambda,\theta)$, $\theta\in\mathbb R^1$, which may have zeros, or, more specifically, $\rm{mes}\{\lambda|f(\lambda,\theta_0)=0\}=0$, where mes denotes the Lebesgue measure. In addition, we prove standard inequalities, the validity of which, along with the property of local asymptotical normality, assures “good” asymptotical properties of the estimates of maximal likelihood and Bayesian estimates for the parameter $\theta$.