Generalization of the Marčenko-Pastur problem
We study the spectrum of generalized Wishart matrices, defined as F=(XY^{⊤}+YX^{⊤})/2T, where X and Y are N×T matrices with zero mean, unit variance independent and identically distributed entries and such that E[X_{it}Y_{jt}]=cδ_{i,j}. The limit c=1 corresponds to the Marčenko-Pastur problem. For a...
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| Published in: | Physical review. E Vol. 102; no. 6-1; p. 062117 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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01.12.2020
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| ISSN: | 2470-0053, 2470-0053 |
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| Abstract | We study the spectrum of generalized Wishart matrices, defined as F=(XY^{⊤}+YX^{⊤})/2T, where X and Y are N×T matrices with zero mean, unit variance independent and identically distributed entries and such that E[X_{it}Y_{jt}]=cδ_{i,j}. The limit c=1 corresponds to the Marčenko-Pastur problem. For a general c, we show that the Stieltjes transform of F is the solution of a cubic equation. In the limit c=0, T≫N, the density of eigenvalues converges to the Wigner semicircle. |
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| AbstractList | We study the spectrum of generalized Wishart matrices, defined as F=(XY^{⊤}+YX^{⊤})/2T, where X and Y are N×T matrices with zero mean, unit variance independent and identically distributed entries and such that E[X_{it}Y_{jt}]=cδ_{i,j}. The limit c=1 corresponds to the Marčenko-Pastur problem. For a general c, we show that the Stieltjes transform of F is the solution of a cubic equation. In the limit c=0, T≫N, the density of eigenvalues converges to the Wigner semicircle. We study the spectrum of generalized Wishart matrices, defined as F=(XY^{⊤}+YX^{⊤})/2T, where X and Y are N×T matrices with zero mean, unit variance independent and identically distributed entries and such that E[X_{it}Y_{jt}]=cδ_{i,j}. The limit c=1 corresponds to the Marčenko-Pastur problem. For a general c, we show that the Stieltjes transform of F is the solution of a cubic equation. In the limit c=0, T≫N, the density of eigenvalues converges to the Wigner semicircle.We study the spectrum of generalized Wishart matrices, defined as F=(XY^{⊤}+YX^{⊤})/2T, where X and Y are N×T matrices with zero mean, unit variance independent and identically distributed entries and such that E[X_{it}Y_{jt}]=cδ_{i,j}. The limit c=1 corresponds to the Marčenko-Pastur problem. For a general c, we show that the Stieltjes transform of F is the solution of a cubic equation. In the limit c=0, T≫N, the density of eigenvalues converges to the Wigner semicircle. |
| Author | Potters, Marc Bouchaud, Jean-Philippe |
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| BackLink | https://www.ncbi.nlm.nih.gov/pubmed/33466081$$D View this record in MEDLINE/PubMed |
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