{"id":{"repo_id":"nus","oai_identifier":"oai:scholarbank.nus.edu.sg:10635/309719"},"canonical_url":"https://search.dev.ndltd.org/etd/nus/oai:scholarbank.nus.edu.sg:10635/309719","repository":{"repo_id":"nus","name":"National University of Singapore","base_url":"https://scholarbank.nus.edu.sg/oai/request"},"display":{"title":"PROBABILISTIC MODELS FOR DEPENDENT VARIABLES: COMPLEX ANALYTIC AND COMBINATORIAL APPROACHES","abstract":"In this thesis, we study probabilistic models for dependent variables using complex analytic and combinatorial approaches. Our study consists of two independent parts. In the first part, we establish central limit theorems (CLTs) under zero-free conditions, building upon a quantitative extension of Marcinkiewicz's Theorem. In particular, we obtain quantitative decay estimates for the Kolmogorov-Smirnov distance between a real-valued random variable X and a Gaussian under the condition that the characteristic function of X does not vanish only in a bounded disk. This leads to a quantitative CLT applicable to very general and possibly strongly dependent random systems. In addition, we generalize our CLT to multi-dimensional cases and propose a variant of this generalization with more relaxed zero-free conditions. In the second part, we investigate an interesting connection between the concepts of vines in probability theory and MAT-labeled graphs in hyperplane arrangement theory. We show that there exists an explicit equivalence between the categories of locally regular vines and MAT-labeled graphs. Several applications will be mentioned to illustrate the interaction between the two concepts. Notably, we give an affirmative answer to a question of Cuntz-Mucksch for the characterization of MAT-freeness in terms of partially ordered sets.","abstract_html":"In this thesis, we study probabilistic models for dependent variables using complex analytic and combinatorial approaches. Our study consists of two independent parts. In the first part, we establish central limit theorems (CLTs) under zero-free conditions, building upon a quantitative extension of Marcinkiewicz's Theorem. In particular, we obtain quantitative decay estimates for the Kolmogorov-Smirnov distance between a real-valued random variable X and a Gaussian under the condition that the characteristic function of X does not vanish only in a bounded disk. This leads to a quantitative CLT applicable to very general and possibly strongly dependent random systems. In addition, we generalize our CLT to multi-dimensional cases and propose a variant of this generalization with more relaxed zero-free conditions. In the second part, we investigate an interesting connection between the concepts of vines in probability theory and MAT-labeled graphs in hyperplane arrangement theory. We show that there exists an explicit equivalence between the categories of locally regular vines and MAT-labeled graphs. Several applications will be mentioned to illustrate the interaction between the two concepts. Notably, we give an affirmative answer to a question of Cuntz-Mucksch for the characterization of MAT-freeness in terms of partially ordered sets.","abstract_has_math":false,"creators":["TRAN MANH HUNG"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2024,"date_issued":"2024-07-03","date_published":"2024-07-03","updated_at":"2026-07-24T03:32:04Z","subjects":["poset","labeled graph","MAT-labeling","vine","central limit theorem","Marcinkiewicz's theorem"],"languages":[],"rights":[],"rights_urls":["https://scholarbank.nus.edu.sg/bitstreams/a9e1d20a-1215-4597-9cb3-2ff5d5b8b47b/download"],"identifier_entries":[]},"links":{"outbound_url":null,"outbound_label":null,"outbound_source":null},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["TRAN MANH HUNG"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2024-07-03"]},{"key":"dc:relation.isreferencedby","label":"Dc Relation Isreferencedby","values":["https://scholarbank.nus.edu.sg/handle/10635/309719"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["poset","labeled graph","MAT-labeling","vine","central limit theorem","Marcinkiewicz's theorem"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["https://scholarbank.nus.edu.sg/bitstreams/a9e1d20a-1215-4597-9cb3-2ff5d5b8b47b/download"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://scholarbank.nus.edu.sg/bitstreams/93266858-739b-4616-a117-48255a82cd23/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["In this thesis, we study probabilistic models for dependent variables using complex analytic and combinatorial approaches. Our study consists of two independent parts. In the first part, we establish central limit theorems (CLTs) under zero-free conditions, building upon a quantitative extension of Marcinkiewicz's Theorem. In particular, we obtain quantitative decay estimates for the Kolmogorov-Smirnov distance between a real-valued random variable X and a Gaussian under the condition that the characteristic function of X does not vanish only in a bounded disk. This leads to a quantitative CLT applicable to very general and possibly strongly dependent random systems. In addition, we generalize our CLT to multi-dimensional cases and propose a variant of this generalization with more relaxed zero-free conditions. In the second part, we investigate an interesting connection between the concepts of vines in probability theory and MAT-labeled graphs in hyperplane arrangement theory. We show that there exists an explicit equivalence between the categories of locally regular vines and MAT-labeled graphs. Several applications will be mentioned to illustrate the interaction between the two concepts. Notably, we give an affirmative answer to a question of Cuntz-Mucksch for the characterization of MAT-freeness in terms of partially ordered sets."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["9f3c6f2aa8f96291ef77b0a18484521d","676c9596e7474af0dd5d45025fa29b7e","6413696e492f995a36e13195ceb34a5c"]},{"key":"dc:title","label":"Title","values":["PROBABILISTIC MODELS FOR DEPENDENT VARIABLES: COMPLEX ANALYTIC AND COMBINATORIAL APPROACHES"]}]}],"canonical_facts":{"dc:creator":["TRAN MANH HUNG"],"dc:date.issued":["2024-07-03"],"dc:description.abstract":["In this thesis, we study probabilistic models for dependent variables using complex analytic and combinatorial approaches. Our study consists of two independent parts. In the first part, we establish central limit theorems (CLTs) under zero-free conditions, building upon a quantitative extension of Marcinkiewicz's Theorem. In particular, we obtain quantitative decay estimates for the Kolmogorov-Smirnov distance between a real-valued random variable X and a Gaussian under the condition that the characteristic function of X does not vanish only in a bounded disk. This leads to a quantitative CLT applicable to very general and possibly strongly dependent random systems. In addition, we generalize our CLT to multi-dimensional cases and propose a variant of this generalization with more relaxed zero-free conditions. In the second part, we investigate an interesting connection between the concepts of vines in probability theory and MAT-labeled graphs in hyperplane arrangement theory. We show that there exists an explicit equivalence between the categories of locally regular vines and MAT-labeled graphs. Several applications will be mentioned to illustrate the interaction between the two concepts. Notably, we give an affirmative answer to a question of Cuntz-Mucksch for the characterization of MAT-freeness in terms of partially ordered sets."],"dc:format.checksum.md5":["9f3c6f2aa8f96291ef77b0a18484521d","676c9596e7474af0dd5d45025fa29b7e","6413696e492f995a36e13195ceb34a5c"],"dc:identifier.uri":["https://scholarbank.nus.edu.sg/bitstreams/93266858-739b-4616-a117-48255a82cd23/download"],"dc:relation.isreferencedby":["https://scholarbank.nus.edu.sg/handle/10635/309719"],"dc:rights":["https://scholarbank.nus.edu.sg/bitstreams/a9e1d20a-1215-4597-9cb3-2ff5d5b8b47b/download"],"dc:subject":["poset","labeled graph","MAT-labeling","vine","central limit theorem","Marcinkiewicz's theorem"],"dc:title":["PROBABILISTIC MODELS FOR DEPENDENT VARIABLES: COMPLEX ANALYTIC AND COMBINATORIAL APPROACHES"],"dc:type":["Thesis"]},"updated_at":"2026-07-24T03:32:04Z"}