ASYMPTOTIC EQUIVALENCE OF LOCALLY STATIONARY PROCESSES AND BIVARIATE GAUSSIAN WHITE NOISE

We consider a general class of statistical experiments, in which an ndimensional centered Gaussian random variable is observed and its covariance matrix is the parameter of interest. The covariance matrix is assumed to be well-approximable in a linear space of lower dimension K_n with eigenvalues uniformly bounded away from zero and infinity. We prove asymptotic equivalence of this experiment and a class of K_n-dimensional Gaussian models with informative expectation in Le Cam’s sense when n tends to infinity and K_n is allowed to increase moderately in n at a polynomial rate. For this purpose we derive a new localization technique for non-i.i.d. data and a novel high-dimensional Central Limit Law in total variation distance. These results are key ingredients to show asymptotic equivalence between the experiments of locally stationary Gaussian time series and a bivariate Wiener process with the log spectral density as its drift. On this way a novel class of matrices is introduced which generalizes circulant Toeplitz matrices traditionally used for strictly stationary time series.