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| Main Authors: | , , , |
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| Format: | Preprint |
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2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2102.05761 |
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| _version_ | 1866909219909795840 |
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| author | Abry, Patrice Boniece, B. Cooper Didier, Gustavo Wendt, Herwig |
| author_facet | Abry, Patrice Boniece, B. Cooper Didier, Gustavo Wendt, Herwig |
| contents | In this paper, we characterize the asymptotic and large scale behavior of the eigenvalues of wavelet random matrices in high dimensions. We assume that possibly non-Gaussian, finite-variance $p$-variate measurements are made of a low-dimensional $r$-variate ($r \ll p$) fractional stochastic process with non-canonical scaling coordinates and in the presence of additive high-dimensional noise. The measurements are correlated both time-wise and between rows. We show that the $r$ largest eigenvalues of the wavelet random matrices, when appropriately rescaled, converge in probability to scale-invariant functions in the high-dimensional limit. By contrast, the remaining $p-r$ eigenvalues remain bounded in probability. Under additional assumptions, we show that the $r$ largest log-eigenvalues of wavelet random matrices exhibit asymptotically Gaussian distributions. The results have direct consequences for statistical inference. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_05761 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On high-dimensional wavelet eigenanalysis Abry, Patrice Boniece, B. Cooper Didier, Gustavo Wendt, Herwig Statistics Theory Probability 60G22, 60B20 In this paper, we characterize the asymptotic and large scale behavior of the eigenvalues of wavelet random matrices in high dimensions. We assume that possibly non-Gaussian, finite-variance $p$-variate measurements are made of a low-dimensional $r$-variate ($r \ll p$) fractional stochastic process with non-canonical scaling coordinates and in the presence of additive high-dimensional noise. The measurements are correlated both time-wise and between rows. We show that the $r$ largest eigenvalues of the wavelet random matrices, when appropriately rescaled, converge in probability to scale-invariant functions in the high-dimensional limit. By contrast, the remaining $p-r$ eigenvalues remain bounded in probability. Under additional assumptions, we show that the $r$ largest log-eigenvalues of wavelet random matrices exhibit asymptotically Gaussian distributions. The results have direct consequences for statistical inference. |
| title | On high-dimensional wavelet eigenanalysis |
| topic | Statistics Theory Probability 60G22, 60B20 |
| url | https://arxiv.org/abs/2102.05761 |