Asymptotic distribution of eigenvalues of random matrices and characterization of the Gaussian distribution by rotational invariance
| dc.contributor.author | Olson, William Howard | en |
| dc.contributor.department | Statistics | en |
| dc.date.accessioned | 2022-05-26T19:30:11Z | en |
| dc.date.available | 2022-05-26T19:30:11Z | en |
| dc.date.issued | 1970 | en |
| dc.description.abstract | The study falls in the area of random equations; in particular properties of random matrices have been studied. The dissertation makes precise some statistical theories of spectra developed in recent years by a number of physicists. Two basic results have been achieved. The first result is a characterization of the distribution of a symmetric random matrix. Assuming independence of the diagonal and super-diagonal random variables of the symmetric random matrix the following theorem is proved: the distribution of the matrix is invariant under orthogonal similarity transforms if and only if the diagonal random variables are normally distributed with mean μ, and variance 2a², and the off-diagonal elements are normally distributed with mean O and variance a², :for some constants μ, and a² > O. The proof is achieved by solving a functional equation in characteristic functions. This seems to have been first proved in this context by Porter and Rosenzweig (Ann. Acad. Sci. Fennicae. AVI, No. 44, 1960) by a different method and under more restrictive conditions than those given here. The second result deals with the asymptotic distribution of eigenvalues of a synnnetric random matrix as the dimension approaches infinity. Let A<sub>n</sub> be an appropriately normalized n ⨉ n symmetric random matrix and let W<sub>n</sub>(x) denote the empirical distribution function of the eigenvalues of A<sub>n</sub. Under suitable conditions on the random variables of the matrix it is proved that W<sub>n</sub>(x)⟶W(x) as n∞, where W is the absolutely continuous distribution function with a semi-circle density, W(x) = { ⎧ 2/π (1-x²)<sup>1/2</sup>, |x| ≤ 1 ⎨ ⎩ 0 , |x| > 1. The proof is achieved by an intricate combinatorial analysis in conjunction with the method of moments. This result extends a conjecture made by E. P. Wigner ("On the Distribution of the Roots of Certain Symmmetric Matrices," Ann. Math. 67, 1958, 325). | en |
| dc.description.degree | Ph. D. | en |
| dc.format.extent | v, 54, 2 leaves | en |
| dc.format.mimetype | application/pdf | en |
| dc.identifier.uri | http://hdl.handle.net/10919/110345 | en |
| dc.language.iso | en | en |
| dc.publisher | Virginia Polytechnic Institute and State University | en |
| dc.relation.isformatof | OCLC# 41155121 | en |
| dc.rights | In Copyright | en |
| dc.rights.uri | http://rightsstatements.org/vocab/InC/1.0/ | en |
| dc.subject.lcc | LD5655.V856 1970.O44 | en |
| dc.title | Asymptotic distribution of eigenvalues of random matrices and characterization of the Gaussian distribution by rotational invariance | en |
| dc.type | Dissertation | en |
| dc.type.dcmitype | Text | en |
| thesis.degree.discipline | Statistics | en |
| thesis.degree.grantor | Virginia Polytechnic Institute and State University | en |
| thesis.degree.level | doctoral | en |
| thesis.degree.name | Ph. D. | en |
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