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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.04371 |
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| _version_ | 1866909909842395136 |
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| author | Loubaton, Philippe Rosuel, Alexis Vallet, Pascal |
| author_facet | Loubaton, Philippe Rosuel, Alexis Vallet, Pascal |
| contents | It is established that the linear spectral statistics (LSS) of the smoothed periodogram estimate of the spectral coherence matrix of a complex Gaussian high-dimensional times series (yn) n$\in$Z with independent components satisfy at each frequency a central limit theorem in the asymptotic regime where the sample size N , the dimension M of the observation, and the smoothing span B both converge towards +$\infty$ in such a way that M = O(N $α$ ) for $α$ < 1 and M B $\rightarrow$ c, c $\in$ (0, 1). It is deduced that two recentered and renormalized versions of the LSS, one based on an average in the frequency domain and the other one based on a sum of squares also in the frequency domain, and both evaluated over a well-chosen frequency grid, also verify a central limit theorem. These two statistics are proposed to test with controlled asymptotic level the hypothesis that the components of y are independent. Numerical simulations assess the performance of the two tests. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_04371 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Correlation tests and sample spectral coherence matrix in the high-dimensional regime Loubaton, Philippe Rosuel, Alexis Vallet, Pascal Statistics Theory It is established that the linear spectral statistics (LSS) of the smoothed periodogram estimate of the spectral coherence matrix of a complex Gaussian high-dimensional times series (yn) n$\in$Z with independent components satisfy at each frequency a central limit theorem in the asymptotic regime where the sample size N , the dimension M of the observation, and the smoothing span B both converge towards +$\infty$ in such a way that M = O(N $α$ ) for $α$ < 1 and M B $\rightarrow$ c, c $\in$ (0, 1). It is deduced that two recentered and renormalized versions of the LSS, one based on an average in the frequency domain and the other one based on a sum of squares also in the frequency domain, and both evaluated over a well-chosen frequency grid, also verify a central limit theorem. These two statistics are proposed to test with controlled asymptotic level the hypothesis that the components of y are independent. Numerical simulations assess the performance of the two tests. |
| title | Correlation tests and sample spectral coherence matrix in the high-dimensional regime |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2501.04371 |