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Stein's Method and the Multivariate CLT for Traces of Powers on the Compact Classical Groups


 
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1. Title Title of document Stein's Method and the Multivariate CLT for Traces of Powers on the Compact Classical Groups
 
2. Creator Author's name, affiliation, country Christian Döbler; Ruhr University Bochum; Germany
 
2. Creator Author's name, affiliation, country Michael Stolz; Ruhr University Bochum; Germany
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) random matrices, compact Lie groups, Haar measure, traces of powers, Stein's method, normal approximation, exchangeable pairs, heat kernel, power sum symmetric polynomials
 
3. Subject Subject classification 15B52; 60F05 ; 60B15; 58J65
 
4. Description Abstract Let $M$ be a random element of the unitary, special orthogonal, or unitary symplectic groups, distributed according to Haar measure. By a classical result of Diaconis and Shahshahani, for large matrix size $n$, the vector of traces of consecutive powers of $M$ tends to a vector of independent (real or complex) Gaussian random variables. Recently, Jason Fulman has demonstrated that for a single power $j$ (which may grow with $n$), a speed of convergence result may be obtained via Stein's method of exchangeable pairs. In this note, we extend Fulman's result to the multivariate central limit theorem for the full vector of traces of powers.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) Deutsche Forschungsgemeinschaft
 
7. Date (YYYY-MM-DD) 2011-11-22
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/960
 
10. Identifier Digital Object Identifier 10.1214/EJP.v16-960
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 16
 
12. Language English=en
 
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