{"id":{"repo_id":"bu","oai_identifier":"oai:open.bu.edu:2144/45267"},"canonical_url":"https://search.dev.ndltd.org/etd/bu/oai:open.bu.edu:2144/45267","repository":{"repo_id":"bu","name":"Boston University","base_url":"https://open.bu.edu/oai/request"},"display":{"title":"On non-stationary Wishart matrices and functional Gaussian approximations in Hilbert spaces","abstract":"This thesis contains two main chapters. The first chapter focuses on the highdimensional asymptotic regimes of correlated Wishart matrices d−1YY^T , where Y is a n×d Gaussian random matrix with correlated and non-stationary entries. We provide quantitative bounds in the Wasserstein distance for the cases of central convergence and non-central convergence, verify such convergences hold in the weak topology of C([a; b]; M_n(R)), and show that our result can be used to prove convergence in expectation of the empirical spectral distributions of the Wishart matrices to the semicircular law. The second chapter develops a version of the Stein-Malliavin method in an infinite-dimensional and non-diffusive Poissonian setting. In particular, we provide quantitative central limit theorems for approximations by non-degenerate Hilbert-valued Gaussian random elements, as well as fourth moment bounds for approximating sequences with finite chaos expansion. We apply our results to the Brownian approximation of Poisson processes in Besov-Liouville spaces and also derive a functional limit theorem for an edge-counting statistic of a random geometric graph.","abstract_html":"This thesis contains two main chapters. The first chapter focuses on the highdimensional asymptotic regimes of correlated Wishart matrices d−1YY^T , where Y is a n×d Gaussian random matrix with correlated and non-stationary entries. We provide quantitative bounds in the Wasserstein distance for the cases of central convergence and non-central convergence, verify such convergences hold in the weak topology of C([a; b]; M_n(R)), and show that our result can be used to prove convergence in expectation of the empirical spectral distributions of the Wishart matrices to the semicircular law. The second chapter develops a version of the Stein-Malliavin method in an infinite-dimensional and non-diffusive Poissonian setting. In particular, we provide quantitative central limit theorems for approximations by non-degenerate Hilbert-valued Gaussian random elements, as well as fourth moment bounds for approximating sequences with finite chaos expansion. We apply our results to the Brownian approximation of Poisson processes in Besov-Liouville spaces and also derive a functional limit theorem for an edge-counting statistic of a random geometric graph.","abstract_has_math":false,"creators":["Dang, Thanh"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Bourguin, Solesne"],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022","date_published":"2022","updated_at":"2026-07-24T01:24:17Z","subjects":["Mathematics","Gamma calculus","Limit theorems","Malliavin-Stein method","Random matrices"],"languages":["en_US"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2144/45267","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Bourguin, Solesne"]},{"key":"dc:creator","label":"Author","values":["Dang, Thanh"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2022-10-26T13:02:23Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2022-10-26T13:02:23Z"]},{"key":"dc:date.issued","label":"Date","values":["2022"]},{"key":"dc:type","label":"Dc Type","values":["Thesis/Dissertation"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics","Gamma calculus","Limit theorems","Malliavin-Stein method","Random matrices"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/2144/45267"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["This thesis contains two main chapters. The first chapter focuses on the highdimensional asymptotic regimes of correlated Wishart matrices d−1YY^T , where Y is a n×d Gaussian random matrix with correlated and non-stationary entries. We provide quantitative bounds in the Wasserstein distance for the cases of central convergence and non-central convergence, verify such convergences hold in the weak topology of C([a; b]; M_n(R)), and show that our result can be used to prove convergence in expectation of the empirical spectral distributions of the Wishart matrices to the semicircular law. The second chapter develops a version of the Stein-Malliavin method in an infinite-dimensional and non-diffusive Poissonian setting. In particular, we provide quantitative central limit theorems for approximations by non-degenerate Hilbert-valued Gaussian random elements, as well as fourth moment bounds for approximating sequences with finite chaos expansion. We apply our results to the Brownian approximation of Poisson processes in Besov-Liouville spaces and also derive a functional limit theorem for an edge-counting statistic of a random geometric graph."]},{"key":"dc:title","label":"Title","values":["On non-stationary Wishart matrices and functional Gaussian approximations in Hilbert spaces"]}]}],"canonical_facts":{"dc:contributor.advisor":["Bourguin, Solesne"],"dc:creator":["Dang, Thanh"],"dc:date.accessioned":["2022-10-26T13:02:23Z"],"dc:date.available":["2022-10-26T13:02:23Z"],"dc:date.issued":["2022"],"dc:description.abstract":["This thesis contains two main chapters. The first chapter focuses on the highdimensional asymptotic regimes of correlated Wishart matrices d−1YY^T , where Y is a n×d Gaussian random matrix with correlated and non-stationary entries. We provide quantitative bounds in the Wasserstein distance for the cases of central convergence and non-central convergence, verify such convergences hold in the weak topology of C([a; b]; M_n(R)), and show that our result can be used to prove convergence in expectation of the empirical spectral distributions of the Wishart matrices to the semicircular law. The second chapter develops a version of the Stein-Malliavin method in an infinite-dimensional and non-diffusive Poissonian setting. In particular, we provide quantitative central limit theorems for approximations by non-degenerate Hilbert-valued Gaussian random elements, as well as fourth moment bounds for approximating sequences with finite chaos expansion. We apply our results to the Brownian approximation of Poisson processes in Besov-Liouville spaces and also derive a functional limit theorem for an edge-counting statistic of a random geometric graph."],"dc:identifier.uri":["https://hdl.handle.net/2144/45267"],"dc:language.iso":["en_US"],"dc:subject":["Mathematics","Gamma calculus","Limit theorems","Malliavin-Stein method","Random matrices"],"dc:title":["On non-stationary Wishart matrices and functional Gaussian approximations in Hilbert spaces"],"dc:type":["Thesis/Dissertation"]},"updated_at":"2026-07-24T01:24:17Z"}