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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.05751 |
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| _version_ | 1866915104930398208 |
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| author | Butucea, Cristina Meister, Alexander Rohde, Angelika |
| author_facet | Butucea, Cristina Meister, Alexander Rohde, Angelika |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_05751 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic Equivalence of Locally Stationary Processes and Bivariate White Noise Butucea, Cristina Meister, Alexander Rohde, Angelika Statistics Theory 62B15, 62M10, 60F05 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. |
| title | Asymptotic Equivalence of Locally Stationary Processes and Bivariate White Noise |
| topic | Statistics Theory 62B15, 62M10, 60F05 |
| url | https://arxiv.org/abs/2410.05751 |