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Distribution of Eigenvalues of Sample Covariance Matrices with Tensor Product Samples
We consider the n² × n² real symmetric and hermitian matrices Mₙ, which are equal to the sum mn of tensor products of the vectors Xμ = B(Yμ ⊗ Yμ), μ = 1, . . . ,mn, where Yμ are i.i.d. random vectors from Rⁿ(Cⁿ) with zero mean and unit variance of components, and B is an n² × n² positive definite no...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2017
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| Series: | Журнал математической физики, анализа, геометрии |
| Online Access: | http://dspace.nbuv.gov.ua/handle/123456789/140566 |
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irk-123456789-1405662018-07-11T01:23:21Z Distribution of Eigenvalues of Sample Covariance Matrices with Tensor Product Samples Tieplova, D. We consider the n² × n² real symmetric and hermitian matrices Mₙ, which are equal to the sum mn of tensor products of the vectors Xμ = B(Yμ ⊗ Yμ), μ = 1, . . . ,mn, where Yμ are i.i.d. random vectors from Rⁿ(Cⁿ) with zero mean and unit variance of components, and B is an n² × n² positive definite non-random matrix. We prove that if mₙ / n² → c ∊ [0,+∞) and the Normalized Counting Measure of eigenvalues of BJB, where J is defined below in (2.6), converges weakly, then the Normalized Counting Measure of eigenvalues of Mn converges weakly in probability to a non-random limit, and its Stieltjes transform can be found from a certain functional equation. 2017 Article Distribution of Eigenvalues of Sample Covariance Matrices with Tensor Product Samples / D. Tieplova // Журнал математической физики, анализа, геометрии. — 2017. — Т. 13, № 1. — С. 82-98. — Бібліогр.: 11 назв. — англ. 1812-9471 DOI: doi.org/10.15407/mag13.01.082 Mathematics Subject Classification 2000: 15B52 http://dspace.nbuv.gov.ua/handle/123456789/140566 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
We consider the n² × n² real symmetric and hermitian matrices Mₙ, which are equal to the sum mn of tensor products of the vectors Xμ = B(Yμ ⊗ Yμ), μ = 1, . . . ,mn, where Yμ are i.i.d. random vectors from Rⁿ(Cⁿ) with zero mean and unit variance of components, and B is an n² × n² positive definite non-random matrix. We prove that if mₙ / n² → c ∊ [0,+∞) and the Normalized Counting Measure of eigenvalues of BJB, where J is defined below in (2.6), converges weakly, then the Normalized Counting Measure of eigenvalues of Mn converges weakly in probability to a non-random limit, and its Stieltjes transform can be found from a certain functional equation. |
| format |
Article |
| author |
Tieplova, D. |
| spellingShingle |
Tieplova, D. Distribution of Eigenvalues of Sample Covariance Matrices with Tensor Product Samples Журнал математической физики, анализа, геометрии |
| author_facet |
Tieplova, D. |
| author_sort |
Tieplova, D. |
| title |
Distribution of Eigenvalues of Sample Covariance Matrices with Tensor Product Samples |
| title_short |
Distribution of Eigenvalues of Sample Covariance Matrices with Tensor Product Samples |
| title_full |
Distribution of Eigenvalues of Sample Covariance Matrices with Tensor Product Samples |
| title_fullStr |
Distribution of Eigenvalues of Sample Covariance Matrices with Tensor Product Samples |
| title_full_unstemmed |
Distribution of Eigenvalues of Sample Covariance Matrices with Tensor Product Samples |
| title_sort |
distribution of eigenvalues of sample covariance matrices with tensor product samples |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2017 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/140566 |
| citation_txt |
Distribution of Eigenvalues of Sample Covariance Matrices with Tensor Product Samples / D. Tieplova // Журнал математической физики, анализа, геометрии. — 2017. — Т. 13, № 1. — С. 82-98. — Бібліогр.: 11 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT tieplovad distributionofeigenvaluesofsamplecovariancematriceswithtensorproductsamples |
| first_indexed |
2023-10-18T21:22:56Z |
| last_indexed |
2023-10-18T21:22:56Z |
| _version_ |
1796152663111368704 |