Asymptotic Equivalence of Locally Stationary Processes and Bivariate White Noise

TL;DR

This paper establishes the asymptotic equivalence between locally stationary Gaussian processes and bivariate Wiener processes, introducing new techniques for high-dimensional data and extending classical models to more general matrix classes.

Contribution

It develops a new localization technique and a high-dimensional CLT, proving asymptotic equivalence of complex Gaussian experiments and Wiener processes with novel matrix classes.

Findings

Asymptotic equivalence between locally stationary Gaussian processes and bivariate Wiener processes.

Introduction of a new localization technique for non-i.i.d. data.

Development of a high-dimensional Central Limit Law in total variation.

Abstract

We consider a general class of statistical experiments, in which an $n$-dimensional 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. Therein a novel class of matrices is introduced which generalizes circulant Toeplitz matrices traditionally used for strictly stationary time series.