{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/110345"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/110345","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Asymptotic distribution of eigenvalues of random matrices and characterization of the Gaussian distribution by rotational invariance","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).","abstract_html":"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 &gt; 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&lt;sub&gt;n&lt;/sub&gt; be an appropriately normalized n ⨉ n symmetric random matrix and let W&lt;sub&gt;n&lt;/sub&gt;(x) denote the empirical distribution function of the eigenvalues of A&lt;sub&gt;n&lt;/sub. Under suitable conditions on the random variables of the matrix it is proved that W&lt;sub&gt;n&lt;/sub&gt;(x)⟶W(x) as n∞, where W is the absolutely continuous distribution function with a semi-circle density, W(x) = { ⎧ 2/π (1-x²)&lt;sup&gt;1/2&lt;/sup&gt;, |x| ≤ 1 ⎨ ⎩ 0 , |x| &gt; 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 (&quot;On the Distribution of the Roots of Certain Symmmetric Matrices,&quot; Ann. Math. 67, 1958, 325).","abstract_has_math":false,"creators":["Olson, William Howard"],"institution":"Virginia Polytechnic Institute and State University","degree_name":"Ph. D.","degree_level":"doctoral","degree_discipline":"Statistics","degree_department":"Statistics","school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1970,"date_issued":"1970","date_published":"1970","updated_at":"2026-07-22T22:19:16Z","subjects":[],"languages":["en"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10919/110345","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.department","label":"Department","values":["Statistics"]},{"key":"dc:creator","label":"Author","values":["Olson, William Howard"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2022-05-26T19:30:11Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2022-05-26T19:30:11Z"]},{"key":"dc:date.issued","label":"Date","values":["1970"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Polytechnic Institute and State University"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"dc:type.dcmitype","label":"Dc Type Dcmitype","values":["Text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Statistics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph. D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/110345"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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)."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Asymptotic distribution of eigenvalues of random matrices and characterization of the Gaussian distribution by rotational invariance"]}]}],"canonical_facts":{"dc:contributor.department":["Statistics"],"dc:creator":["Olson, William Howard"],"dc:date.accessioned":["2022-05-26T19:30:11Z"],"dc:date.available":["2022-05-26T19:30:11Z"],"dc:date.issued":["1970"],"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)."],"dc:description.degree":["Ph. D."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/10919/110345"],"dc:language.iso":["en"],"dc:publisher":["Virginia Polytechnic Institute and State University"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["Asymptotic distribution of eigenvalues of random matrices and characterization of the Gaussian distribution by rotational invariance"],"dc:type":["Dissertation"],"dc:type.dcmitype":["Text"],"thesis:degree_discipline":["Statistics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Ph. D."],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:19:16Z"}