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). / Ph. D.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/110345 |
| Date | January 1970 |
| Creators | Olson, William Howard |
| Contributors | Statistics |
| Publisher | Virginia Polytechnic Institute and State University |
| Source Sets | Virginia Tech Theses and Dissertation |
| Language | English |
| Detected Language | English |
| Type | Dissertation, Text |
| Format | v, 54, 2 leaves, application/pdf, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
| Relation | OCLC# 41155121 |
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