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...
Gespeichert in:
| Datum: | 2017 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2017
|
| Schriftenreihe: | Журнал математической физики, анализа, геометрии |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/140566 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | 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. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| Zusammenfassung: | 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. |
|---|