Simultaneous Inference for Covariance and Precision Matrices of Long-Range Dependent Time Series
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| Format: | Preprint |
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2026
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| _version_ | 1866914483794870272 |
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| author | Zhai, Percy S. Kolar, Mladen Wu, Wei Biao |
| author_facet | Zhai, Percy S. Kolar, Mladen Wu, Wei Biao |
| contents | For time series with long-range temporal dependence, inference for covariance and precision matrices is non-trivial. We propose a Berry-Esseen type Gaussian approximation result that gives a finite-sample bound for the Kolmogorov distance between the infinity norms of the estimation error of sample covariance matrix and the corresponding Gaussian approximation. The method utilizes martingale and m-dependent approximation and relies on constructing triadic blocks. We also establish a bootstrapping result with block sampling method, which preserves validity despite strong temporal dependence. Our results on covariance allow ultra-high-dimensional settings where the dimension of time series can grow sub-exponentially with sample size. Similar results can be built for precision matrix under low-dimensional settings. No assumption is required on the structure of covariance and precision matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_16219 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Simultaneous Inference for Covariance and Precision Matrices of Long-Range Dependent Time Series Zhai, Percy S. Kolar, Mladen Wu, Wei Biao Statistics Theory Methodology For time series with long-range temporal dependence, inference for covariance and precision matrices is non-trivial. We propose a Berry-Esseen type Gaussian approximation result that gives a finite-sample bound for the Kolmogorov distance between the infinity norms of the estimation error of sample covariance matrix and the corresponding Gaussian approximation. The method utilizes martingale and m-dependent approximation and relies on constructing triadic blocks. We also establish a bootstrapping result with block sampling method, which preserves validity despite strong temporal dependence. Our results on covariance allow ultra-high-dimensional settings where the dimension of time series can grow sub-exponentially with sample size. Similar results can be built for precision matrix under low-dimensional settings. No assumption is required on the structure of covariance and precision matrices. |
| title | Simultaneous Inference for Covariance and Precision Matrices of Long-Range Dependent Time Series |
| topic | Statistics Theory Methodology |
| url | https://arxiv.org/abs/2604.16219 |