On a Generalisation of the Marcenko-Pastur Problem
We study the spectrum of generalized Wishart matrices, defined as \(\mathbf{F}=( X Y^\top + Y X^\top)/2T\), where \(X\) and \(Y\) are \(N \times T\) matrices with zero mean, unit variance IID entries and such that \(\mathbb{E}[X_{it} Y_{jt}]=c \delta_{i,j}\). The limit \(c=1\) corresponds to the Mar...
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| Published in: | arXiv.org |
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| Main Authors: | , |
| Format: | Paper |
| Language: | English |
| Published: |
Ithaca
Cornell University Library, arXiv.org
20.09.2020
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| Subjects: | |
| ISSN: | 2331-8422 |
| Online Access: | Get full text |
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| Summary: | We study the spectrum of generalized Wishart matrices, defined as \(\mathbf{F}=( X Y^\top + Y X^\top)/2T\), where \(X\) and \(Y\) are \(N \times T\) matrices with zero mean, unit variance IID entries and such that \(\mathbb{E}[X_{it} Y_{jt}]=c \delta_{i,j}\). The limit \(c=1\) corresponds to the Marcenko-Pastur problem. For a general \(c\), we show that the Stietjes transform of \(\mathbf{F}\) is the solution of a cubic equation. In the limit \(c=0\), \(T \gg N\) the density of eigenvalues converges to the Wigner semi-circle. |
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| Bibliography: | SourceType-Working Papers-1 ObjectType-Working Paper/Pre-Print-1 content type line 50 |
| ISSN: | 2331-8422 |
| DOI: | 10.48550/arxiv.2009.07113 |