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Wigner theorems for random matrices with dependent entries: Ensembles associated to symmetric spaces and sample covariance matrices


 
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1. Title Title of document Wigner theorems for random matrices with dependent entries: Ensembles associated to symmetric spaces and sample covariance matrices
 
2. Creator Author's name, affiliation, country Katrin Hofmann-Credner; Ruhr University Bochum
 
2. Creator Author's name, affiliation, country Michael Stolz; Ruhr University Bochum
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) random matrices, symmetric spaces, semicircle law, Wigner, Marcenko-Pastur, Wishart, sample covariance matrices, dependent random variables, density of states, universality
 
3. Subject Subject classification Primary 15A52, Secondary 82B44
 
4. Description Abstract It is a classical result of Wigner that for an hermitian matrix with independent entries on and above the diagonal, the mean empirical eigenvalue distribution converges weakly to the semicircle law as matrix size tends to infinity. In this paper, we prove analogs of Wigner's theorem for random matrices taken from all infinitesimal versions of classical symmetric spaces. This is a class of models which contains those studied by Wigner and Dyson, along with seven others arising in condensed matter physics. Like Wigner's, our results are universal in that they only depend on certain assumptions about the moments of the matrix entries, but not on the specifics of their distributions. What is more, we allow for a certain amount of dependence among the matrix entries, in the spirit of a recent generalization of Wigner's theorem, due to Schenker and Schulz-Baldes. As a byproduct, we obtain a universality result for sample covariance matrices with dependent entries.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) Deutsche Forschungsgemeinschaft
 
7. Date (YYYY-MM-DD) 2008-07-01
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/1395
 
10. Identifier Digital Object Identifier 10.1214/ECP.v13-1395
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 13
 
12. Language English=en
 
14. Coverage Geo-spatial location, chronological period, research sample (gender, age, etc.)
 
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